Fina 


$C 165 752 


IMPRIMITIVE SUBSTITUTION GKOUPS 
OF DEGREE SIXTEEN 


. DISSERTATION 


PRESENTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE 
DEGREE OF DOCTOR OF PHILOSOPHY IN THE GRADUATE 
SCHOOL OF THE OHIO STATE UNIVERSITY. 


By 
Grace MarI£ BAREIS 


THE OHIO STATE UNIVERSITY 
1909 


# 


Sinthsiies endeetatien ee eer 
Lists of groups i ae Si. 
1. Tao aysieine Gbimpealieee = BOs has bea ete 


2. Four systems of imprimitivity.................... 


3. Hight systems of imprimitivity..... 2... .sesee. 


O00 
Pikes 


* 


IMPRIMITIVE SUBSTITUTION GROUPS OF 
DEGREE SIXTEEN. 


Sixteen symbols can be divided in three ways into sets. containing an 
equal number of symbols, viz., into two 8’s; four 4’s, or eight 2’s. In 
the following fist a group isc ssed undes the firs+ systems of int 
to which its head belongs Yor example a gre-~: havi 
four systems of imprimiti»iey is listed amor e 

In previous lists Jetters of the alphabet have usualy” rarer 
elinents. n this list letters’ With subscripts are used, the letters to 
distinguish between systems and the numbers as subscripts to dis- 
tinguish between elements within the system. The sixteen symbols 
used differ therefore with the number of systems. 

For two systems of imprimitivity they are 


Cy Up Ug My Us Ug G7 Og by by bg by bs be br bs 
for four systems they are 


Ay Ap Az Ay by be bg by 1 C2 €3 C4 dy de dg dy 
for eight systems they are 


ay Az by de ey C2 dy dz ey eo fife gi 92 hy he 


The transformations necessary for the final comparison of groups of 
different systems are very simple. 

The transitive constituents of the heads are distinguished by number- 
ing them as they are numbered by Professor G. A. Miller in his “ Memoir 
on the Substitution Groups whose Degree does not Exceed Eight” 
[American Journal of Mathematics, vol. 21 (1899)]. For example 
* (@120344454g474s)32,, is the eleventh group of order 32 and degree 8 in 
Professor Miller’s list. Each distinct head is written and is followed by 
substitutions which generate groups that multiplied into the given head 
produce distinct imprimitive groups. In this list the following notation 
is used: 

(1) Two systems, 


= ayh;. aaby. agbs. ash, . Asb5. agbg. Azb7. aghs 
3 


492895 


4 GRACE MARIE BAREIS 


(2) Four systems, 


t = aybi. debe. agbs. aabs. Crd. Code. esds. cad 
t = aybyey. agbece. agbsc3. aabace 
ts = ab; ‘ dabs ‘ Agbs ; agbs 


Of these ¢ t; generate a group simply isomorphic to the alternating group 
of degree four and ¢ t; ts generate one simply isomorphic to the symmetric 
group of degree four. 

(3) Eight systems, 


t = arbrerdierfign . dabrCodoeofoge 

be = merfidibrergihy . dreofodebocogohe 

tz = didieigrerfi. bodacegreofe 

ty = Iya. didy. e191. erf1. hode. bode. Coge . eo fo 
ts = bigidicr. er fi. dogodoce. afo 

ts = aybyhy. aabohe 

ty = ayhy. ache 


Of these t; t7 generate a group simply isomorphic to the symmetric group 
of degree eight; t; %& to the alternating group; t; & t;” 4 to the primitive 
group of order 1344; t & ts to the 336; t h? ts? to 1681; & t& t” to 1682; 4 t 
to the 56 group. . 

The generators of the tails involve only the t’s, or they involve sub- 
stitutions that do not interchange the systems multiplied by ?¢’s. To 
illustrate 


[ (araeasa4A5AgQ7As) 64, (Dibobsbsbsbebzbs) 64, be, got, Ay Ag2Aq. Dibsbaby .t 


are the two groups that have for their common head a 32, 32 isomorphism 
of the fourth group of order 64 in Professor Miller’s list, and for tails 
this head multiplied by ¢ and ajagaeag.bibsbebs.t respectively. This 
notation is still further abridged to 
[ (ai2s45678)64, (Dias4s678)64, ls2, 32 t,  rsea- Digog.t 

Similarly 

[ (ar2s4)4(Di2za)4(Croa)4(dizza)a | tt ts, tt die. bie. Cre. dia. ts 
are the two groups having for their common head the direct product of 
the non-cyclic transitive groups of degree four and having for tails this 
head multiplied by each of the 23 substitutions different from identity 
of the groups generated by ¢  t and t t; die. bis. cio. diz. ts respectively. 


Systems or INTRANSITIVITY oF Heaps 8, 8. 


Order No. 


3251404800 1 : [ (ar2345678) all (Di2345678) all] t 
1625702400 1 ’ 2 [ (dy2345678) all (bi2345678) all ] pos t, Ayo. 
812851200 1 [ (a12345678) POS (Di2345673) pos | t 
3612672 1 [ (ai2s45678)1344(Diosase7s) 1344] t 
2654208 As [(ai2ssse78)1152(Diesase7s)1152_] ¢ 
1327104 1,2 [ (@r2345678) 1152 (Di2s4se78) 1152] dim t, d15.26.37-48-¢ 
3,4 [ (@izsase7s)1152(Dies4s678)1152_ls76, 576 t, Gio. t 
5, 6 [ (aresase78)1152(Di2s4s678)1152_)576, 576 t, dig.t 
1,4 [ (ai2345678) 1152 (Di2s45678)1152_bss, 288 t, diz-t, 
(45.26-37-48-¢, A1625-37-48-E 
5 [ (Gieg4s67s) 1152 (b12345078)1152_bss, 288 ¢ 
- 6 [ (a12345678)1152 (12345678) 1152_bss, 288 F 
7 [(au2a4s678) 1152 (D12345678)1152_hss, 288 t 
8 [ (aies4s678)576,(Dios4s678)576,_] € 
9 [ (12845078) 576, (b12945678)576,_| ¢ 
1,2 [(Giessse7s)1152(Diosiso7s) 1152144, 144 t, Ar2.56.t 
3,4 [ (a2s45678) 1152 (Diog4se7s) 1152_h44, 144 t, Gia-56.t 
5 {_(G12845678)576,(D12345678) 576, less, 288 15-26-37-48 -£ 
6,9 * [(ai2sss67s)576,(bi2315678)576, bss, 288 t, Aiz-s6-t, 
Ay. byt, Ase. bia. t 
10 [ (aies45678)576, (D12945678)570,_b88, 288 1625-37-48 ¢ 
294912 1 [ (aies4567s)384(Diossse7s)se4_] t 
225792 1 [ (@s2545678)336(Di2s4s678) 336 _] ¢ 
165888 1,4 [ (ai2845678)576,(Di2s45678)576, has, 144 t, Gie-s6-t, 


45-26-3748 - t, ae. Dye a 


663552 


331776 


5 [ (aies45678)576,(Dy2345678) 576, has, 144 ¢ 
6, 7 [ (ar2345678)576, (Di2ss5678) 576, 44, 144 1625-37-48 - t, 
s6. Dip .t 


8 [ (ai2s15678)288(Dr2sis678)288_] ¢ 
1,2 [ (ai2345678)384(Dresas678)3s4_ho2, 192 t, Ais.24-t 
3,4 [ (ai2345678)384(Dr2s4s678)384_hoe, 192 t, ie. 


3) 


147456 


Grace Marre BAREIS 


Order No. 


5, 6 [(ar2345678) 384 (Di2s4s678)384 hoe, 192 t, Aie.t 
112896 Pe [_ (12345678) 336(D12845678)336_hes, 168 t, Cised0378-t 
110592 1 [ (a12345678)576,(D12345678)576, 196, 96 f 
82944 15.2 [ (ar2s45678)288(Dros4s678)oss_la4, 144 t, A15.26-37-48-t 
80640 1 [ (a1e34s678) all (b12345678) all hi, it 
73728 1 [ (a12345678) 1925 (Dr2345678)192,_| ¢ 
2,95 [ (a12845678) 384 (Diosase7s)as4_lo6, 96 t, Gi2-t, Ar3.24.t, 
(493 .t 
6 [ (a12345678)384(Di2345678)384_lo6, 96 t 
7 [ (ai2345678) 384 (Di2sase78)384_lo6, 96 ¢ 
8 [ (a12345678) 384 (Dies4s678)384_lo6, 96 ¢ 
9 L (a12345678)192, (D12845678)192, | # 
10 [ (i2345678) 192, (Diosase7s)192, | ¢ 


11 [_(ar2s45678) 192, (Di2sis678)192, |'t 
56448 1 [ (a12845678) 168, (D12345678)168,_] ¢ 
2 [ (a12345678)168,(D12345678) 168, | ¢ 
55296 1,2 L (ar2s45678) 5765 (Dr2345678)576,_l4s, 48 t, G16.25-37-48-¢ 
3, 4 [ (a12845678)576,(012345678)576,4s, 48 t, 16.25-37-48-t 


49152 1 [ (Giosas67s)384(Dies4s678)384_les, 64 ¢ 
40320 i 2 [ (ar2345678) pos (b12345678) pos h, 1 t, Ay. dy.t : 
36864 1 [ (dies45678) 1152 (b.2345678) 1152eh6, 16 t 


2 [ (d12345678)192,(Di2345678) 1925 |96, 96 15-26-37-48-¢ 
3,4 [ (a12845678)192, (012345678) 1925 96, 96 t, A12.56.¢ 
5, 6 [ (aiz4s678)192,(D12845678) 1925 lo6, 96 t, A15.26-37-48.¢ 
7,8 [ (a12345678)192, (Di2345678)192, 96, go t, Aia.t 
9, 10 [ (ai2sa5678)192, (b12345678) 192, lo6, 96 t, Gig.24.¢ 


11,12 — [(aiesss678)192, (12345678) 192, los, 96 t, Gr423.t 
32768 1 [ (a2345678) 38 (Di2s4s678) 128_| f 
27648 
24576 


1,2 [ (aizga5678)288(Dr2345678)288 4s, 48 t, A12-56- Di2-56.¢ 

1,2 [ (ai2845678)384(D12345678)384_[32, 32 t, G1o.¢ 

3, 4 [ (Gizgas678)384 (12345678) 3842, 32 t, Aio.t 

5,7 [ (a12345678) 192, (D12345678)192,_le4, 64 t, 135-246. ¢, 
133.24. Dig.o4.t 


21504 1 [ (ar2sas678) 1344 (Die315678)1344_ls, 8 t 


IMPRIMITIVE SUBSTITUTION GrouUPS OF DEGREE SIXTEEN 


Order 
18432 


16384 


12288 


9216 


8192 


8, 11 


[ (arosss67s)576,(D12315078)576, he, 16 t, 12. Di2.t 

[ (ai2845678)576,(D12345678)576, he, 16 t, 12. die.t 

[ (a12845678)192,(D12345678)192, hs, 48 15-26-37-48- f, 
16.25-37-48« t 

[(Gies4s678)192,(D12345678)192, xs, 48 t 

[(ai2s45678)192,(D123156789)192, hs, 48 t 

[ (aizsase7s) 965 (012845678) 96,9_] t 

[ (a2s45e78) 96, (Pr2s4s678) 96, _] 4 

(12845678) 96, (D12345678) 96, ,_] t 

[ (a12345678) 128 (Diosss678)128 los, 64 t, G15.26-37-48. 6 

[(ai2ss5678)128(Drosiso7s)128 Jo, 64 t, Gio. t 

[ (ai2s15678)128(Diosiso7s)128 Jos, 64 t, Cie .t 

[(ai2s45678)128(Di2sis67s)128 Jos, 64 t, Giz.24.¢ 

[ (ar234567)128(Di2saso7s)128 lo, 64 t, Aig.24.¢ 

[ (Gr2s45678)128(Diosis67s)128 les, 64 t, Giz. t 

[ (@ies4s678)128(Diez4s678) 128 les, 64 t, Cio. t 

[(ai2345678)192, (12345678) 192, be, 32 t 

[ (12345678) 384 (D12345678)384 he, 16 t 

[ (ar2345678)192, (Di2345678)192, be, a2 t, diz.t, 
133.04. Dy3.04.4, i403. Dig.24.t 

[ (ai2315678)192, (Di2sis678)192, hse, 32 ¢ 

[ (ai2845678)192, (Di2s45678) 192, 32, 32 f 

[ (ai2s15678)288(Di2sss678)288_he, 16 t, ies-s67-t, 
ay. bis ° t, 2-56 + Dio.56. t 

[ (a2845678) 96,, (012345678) 96,_] 215-26-37-48  £ 

[ (ai2345678) 96, (b12345678) 9¢,,_] @15-26-37-48 - ¢ 

[ (aies45678) 128 (Diosss67s) 128 be, 32 t, diet, 
45.26-37-48-€, Ai925-37-48- 

[ (ar2ss5678) 128 (Dizsss678)128 Jee, 32 

[ (12345678) 128 (Diosase7s)128 be, 32 t 

[ (as2345678) 128 (Diosase78) 128 be, 32 f 

[ (aiesss67s)128(Dioss67s)128_)s2, 32 t, ig.o4.t 
45.26.37-48-, 1735-2846. 


[ (12345678) 128 (Dizs4s678) 128 hse, 32 t 


Order 


6272 


No. 
13 
14 

15, 18 


19 
20 
21 

22, 25 


26 

27 

28 
29, 32 


33 

34 

30 
36, 39 


40 

4] 

42 
43, 46 


47 
48 
49 
50 
51 
52 
53 
54 
5d 
1 


Grace Marie BarEIs 


[ (ar2s4s678) 128 (Di2g4s678)128 2, 32 f 

[ (aizs45678) 128 (D2s4s678)128 ke, 32 ¢ 

[ (aros45678) 128 (D12s45678)128 2, 32 t, 
(145.26-37-48-£, C1625-37-48-¢ 

[ (ar2s45678)128 (Dies45678)128 2, 32 F 

[ (ai2sas678) 128 (Diz3s678)128 kee, 32 t 

[ (2345678) 128 (D12245678) 128 hse, 32 f 

[ (ar2345678)128(D12845678)128 2, 32 t, 

Agog. 

[ (ai2sa5678)128 (D12345678) 128 hse, 32 f 

[ (a12345678) 128 (Die345678) 128 se, 32 f 

[ (a1345678)128(Dies45678)128 hse, 32 t 

[ (ar2345678) 128 (Dies45678)128 be, 32 t, 
(324. t 

[ (av345678)128 (Diesa5678) 128 32, 32 F 

[ (ai23a5678) 128 (Di2g4s678) 128 hse, 32 ¢ 

[ (a12345678) 128 (D12g45678)128 hse, 32 f 

[ (c12345678)128 (D12345678) 128.32, 32 t, 
(11324 « t 

[ (ai2345678) 128 (Die345678)128 |s2, 32 ¢ 

[ (ar2345678) 128 (Diesis678)128 hse, 32 ¢ 

[ (c12345678)128 (D12345678) 1282, 32 t 

[ (a12345678)128 (Dies45678)128 be, 32 f, 
14324. t 

[ (ai2s45678) 128 (D12345678) 128 be, 32 t 

[ (aizga567s)128 (D12s4s678)128_he, 92 

[ (ar2345678)128 (Di2345678)128_|s2, 32 F 

[ (aizsase7s) 64, (Dizsase7s) 64, | # 

[ (aizsase78) 64, (Dizsase7s)64,_| ¢ 

[ (ai2345678) 64, (D12845678) 644 | t 

[ (aress5678)64, (Dizsase73)64,_| t 

[ (aizs4s678) 64, (Dr2345678) 64, | t 

[ (aies45678) 64, (b12345678)64,4_| t 

[ (Gies45678) 56 (Diosase7s) s6_| t 


ay. 


ay. 


ay. 


a2 


a2 


‘, 43-24 + t, 


t, 143.24. t, 


Pe t, 143.24. t, 


° t, Q13.24.F, 


ee 


IMPRIMITIVE SUBSTITUTION GROUPS OF DEGREE SIXTEEN 


Order 
6144 


[ (ar2s4s678)192,(Di2sss678)192, his, 16 t, G15.26-37-48 -¢ 
[ (ai2845678)192,(Di2345678)192, he, 16 t, @15-26-87-43-t 


[ (a12345678) 96, (012845678) 964 h2, 32 t, Gi2s-567-t, 


10.56 - D19.56.t 


[ (a12845678)384(Diesss678)384_]s, 8 t, iat 

[ (ai2sss678)384(Diesise7s)ss1_]s, st, iz t 

[ (a2s4s678) 192, (Di2sase7s)192, 18, 8 t, dis.24- Dis.204.¢ 
[ (ai2845678) 96,4 (012345678) 96,182, 32 t, 135-246. t, 


143.24. big.04 By 


[ (ai2345678)48;5(D12345678)48,,_] ¢ 
[(arosss678)48,, (Drosss678)48,,_] t 


[ (a12345678)128(Diesss678)128_ his = 
L (aizs4s67s)128 (Dizsss678)128 ho, 
[ (a2s45678)128 (biess5678)128 his, 
[ (a12345678) 128 (Di2s45678) 128 hs, 
[ (@12345678)128 (Di2s45678)128_he 
[ (ai2s45678) 128 (Dios4se78)128_he, 
[ (Gi2345678)128 (Dios4se78)128_hs, 

115.26-37-48-t, 1625-37-48 ¢, 

Ay 7362845 « t . 
[ (ai2ss5678)128 (D12315678)128._he ; 
[ (a12345678) 128 (D12345678) 128 hs, 
[ (ai2345678) 128 (Di2s4s678)128_he, 
[ (ai2345678)128(Diesss678)128 hs, 
[ (12345678) 128 (2345678) 128 he : 
[ (Gi2s4s678)128 (Di2345678)128 hs, 
[ (ai2s45678) 128 (Di2s4s678) 128 hs 
[ (aies45678)128 (Di2sis678)128_hs ‘ 
[ (a12345678)128 (Dr2348678)128_hs : 
[ (a2345678) 128 (Di2s4s678)128_h, 
[ (@i2a45678)128(Di2845678) 128 hs ; 


[ (ai2545678)128 (Di2345678) 128 hs, 
[ (ai2s45678) 128 (Di2345678)128_hs, 
[ (ar2s45678)128 (12345678) 128_hie 


16 t, 
16 t, 
16 t, 
16 t, 
16 t, 
16 t, 
16 t, 


42.56 + t 

112.56 + t 
143.24.57-48-¢ 
13.24.57-68 « t 
112.56 + t 

142.56 « t 

ape. t, 113.24+ t, 


11735-2846 - t, 11324 - t, 


16 13-24. t 


16 t, 
16 t, 
16 t, 
16 t, 
16 t, 
16 t, 
16 t, 
1 t, 
16 t, 
16 t, 
16 t, 
16 t, 
1 t, 


15.26.37-48-¢ 
1735-2846 - t 
ae. t 

Aye. t 

ay. t 
45.26-37-48-t 
115.26-37-48-¢ 
4625-37-48 - t 
15.26-37-48 -¢ 
1625-37-48 + t 
13-24. t 

4 324- t 
(13-24- t 


9 


10 


Order 


No. 
48, 49 
50, 51 
§2;'53 
54, 55 
56, 57 
58, 59 
60, 61 
62, 63 
64, 65 
66, 67 
68, 69 
70, 71 
12; 3 
74, 75 
76, 77 
78, 79 
80, 81 
82, 83 
84, 85 
86, 87 
88, 89 
90, 91 
92, 93 
94, 95 
96, 97 
98, 99 
100, 101 
102, 103 
104, 105 
106, 107 

1 

2.5 


6 


GracE Marre BAREIS 


[ (ai2345678)128 (D12345678)128_he, 16 t, 


. [ (a12345678) 128 (Dies45678) 128 he, 16 t, 


[ (ar2sas678) 128 (D12sas678)128_he, 16 #, 
[ (12345678) 128 (D12sas678)128_his, 16 
[ (a1o345678) 128 (D12s45678)128_hs, 16 F, 
[ (arogas678)128 (Diesase78) 128 his, 16 ¢, 
[ (a12345678)128 (D12345678)128_he, 16 t, 
[ (arzs45678) 128 (b12845678)128 he, 16 ¢, 
[ (@12345678) 64, (b12345678)64,_|s2, 32 t, 
[ (aiosase78) 64, (Di2345678) 64, 132, 32 t, 
[ (a12345678) 64, (D12945678) 64, 2, 32 t, 
[ (a1osase78) 64, (O12345678) 645.132, 32 t, 


é [ (ai2s4s678) 64, (D12815678) 645 2, 32 F, 


[ (aizsase7s) 64, (Di2sase78) 64, ke, 32 t, 
[ (aiosase78) 64, (D12345678) 645 2, 32 F, 
[ (d12345678) 645 (by2345678) 64, _b2, 32 t, 
[ (ai2345678) 64, (012345678) 64, 2, 32 ¢, 
[ (a12345678) 64, (012345678) 64,_e, 32 t, 
[ (ai2345678) 64, (bio345678) 64, 132, 32 t, 
[ (ai2s4s678) 64, (Diosase7s)64,_hs2, 32 F, 
[ (aiasas678) 64, (Di2345678)64,_h2, 32 #, 
[ (ai2sas678) 64, (bi2345678)64,_h2, 32 #, 
[ (a12345678) 64, (D12345678) 64, be, 92 ¢, 
[ (a2s45678) 64, (Drvass678) 64. foe, a2 t, 
[ (ai2345678) 64, (D12345678) 64, 2, 32 F, 
[ (ar2345678) 645 (012845678) 645 32, 32 t, 
[(a12345678) 645 (D12345678) 64h, 32 t, 
[ (aiosas678)64, (Di2s45678) 64, be, 32 ¢, 
[ (aros45678) 64, (12345678) 64, 2, 32 ¢, 
[ (aros4se78)64, (Diosase7s) 64, bse, 22 #, 
[ (aios45678)192, (D12345678)192, Is, 8 t 
[ (a12345678)192, (Di2ss5678)i02, Is, 8 t, 
(113.24. Di3.04.t, Cisps- by3.24. 
[ (a12345678)192, (Diesase78)192, 18, 8 t 


14324» t 
143-24 t 
324.¢ 
143-24- t 
14324. t 
(113.24 - t 
14324» t 
M2. t 
(145.26-37-48 « 
143-24 - t 
Ay3.24.¢ 
Ay2.56 .¢ 
42-56 + t 
1625-37-48+¢ 
142.56 + t 
Ai3.24.t 
(145.26-37-48 + € 
dye. t 
ay. t 
1735-2846 - t 
ay. t 
ay. t 
(145.26-37-48 +t 
14324 » t 
14324 - t 
142.56 - t 
(142.56 t 
4625-37-48 « t 
14324 - t 
Ay324..t 


Ay. t, 


IMPRIMITIVE SUBSTITUTION GROUPS OF DEGREE SIXTEEN 11 


Order 


z 


2 


_ 
or 


an 


“ 


“ 


“ION 09 
CO > 


Ra) 
sp WDNIAONhWH 


— ht 
oe 


[ (ai2345678)192, (Di2sss678)192, 18, 8 ¢ 

[ (a12845678)192, (Di245678)192, 18, 8 t 

[ (a12345678)192, (Di2sss67s)192, 18, 8 ¢ 

[ (a12345678) 96,5 (12345678) 96,06, 16 t, Aies-567-t, 
1256+ Di2.56.t, 15.26-37-48-¢, A16.25-37-48 - Di2.56.¢ 


[ (12345078) 96,, (b12345678) 96,1 hs, 16 

[ (ar2845678)1344(Dios4s678)1344_h, 1 t 

[ (aiessse7s)168, (P12345678)168, 18, 8 t 

[ (au2s4s678)1152(Diosase7s)1152_h, 1 t 

[ (ai2345678)48,,(Or2345678)48,,_b4, 24 15-26-37-48-¢ 

[ (@ie45678)48,,(O12345678)48,, ba, 24 t, Aro-56-t 

aaa SEC eg 24t, die.56.t 

[ (a12845678)48,,(D12345678)48,, bs, 24 t, Gis.24.73.t 

[ (a12845678)32,, (012345678) 3249_] 

['(a12345678)32,  (Di2sas678)32,4_1 f 

[ (ar2345678)32,,(Di2345678)32,, | ¢ 

[ (ai2s4s678)32,, (D12sas678)32,,_] t 

[ (ar2s4s678)s2,,(Di2s45678)32,, | ¢ 

[ (ai2345678) 32,5 (Diesss678)32,,_| t 

[ (ax2345678) 321, (D12345678)32,,_] ¢ 

[ (a12345678)32,,(Di2345678) 32,6_] ¢ 

[ (ar2sas678)64, (Di2sise7s)64, he, 16 t, Ar3-24.t, 
15.26-37-48-t, 1735-2846. ¢ 

[ (ar2s4s678) 64, (D12845678)64, his, 16 f 

[ (a12345678)64, (D12s45678)64,_h16, 16 ¢ 


[ (ar2s45678)64, (Di2sas678)64, 6, 16 t 


[ (diesss67s)64, (Di2sas678)64, he, 16 t, Ar3.24., 
: Ay2.56-F, 304.56 -¢ 

[(di254s078) 64, (Di2s4s678)64, ho, 16 ¢ 

[ (ai2s4s678)64, (Dizsis678) 64, hs, 16 t 

[ (a12345678) 64, (D12s45678)64, he, 16 t 

[(aiosss67s)64, (Di2ssse7s)e4, he, 16 t, Ais.24-t, 


12.56 + t, 1324-56 - t 


[ (Giesss678)64, (Di2ssse7s)04,_h6, 16 t 


12 


Order 


50, 53 
54, 57 
58, 61 
62 
63 
64 
65, 68 


69 
70, 71 


72, 75 
76 


77 
78 


Grace Marre BAREIS 


[ (arosase7s)64, (D12345678) 645 h6, 
[ (aies45678) 64, (b12845678)64,_he 3 


[ (Gizsas678) 64, (bie345678)64, hs, 
y2.56-t, Q15.26-37-48-¢ 


[ (ai2345678)64, (Di2sase78)64,_he, 
[ (aizs45678) 64, (D12845678) 645 he, 


Ayo.56-t, €15.26-37-48- ¢ 


L (12345678) 64, (biogase78)64,_he, 
[ (a12gas678) 64, (D12345678) 64, he, 


a2. by * t, 56 - by. t 


[ (aiesase7s)64, (Drosase7s)o4, he, 


(1625-37-48, Cr7362845-t 


[ (a12345678) 645 (bies4s678) 64, hs, 
[ (dies45678) 64, (bi2sas678)64,_he, 
[ (a12345678) 64, (bi2345678)64,_he, 


[ (aiesas678)64, (D12345678)64,_h6, 
yy. by2.t, Aige4.Die.t 


[ (a12345678) 64, (D12345678) 644 he, 
A15.26-37-48-t, 57-63 613.04. 
[ (12345678) 64, (D12345678)64,_he, 


15-26-37-48+¢, 1625-37-48 t 


[ (@izs45678) 64, (bi2s45678)64,_he, 
[ (aiesa5678) 64, (biezase78) 64, he 5 
[ (ai2345678) 64, (bio345678)64, hs, 
[ (G12345678) 64, (Dizsase7s)64,_h6 * 

G15:26-37-48-t, Gig-24- D13.24. 
[ (12845678) 64, (Dios4s678)646_he, 
[ (a1os45678) 64, (Die345678) 64, hs, 


(157.68 - D13.24.¢ 


[ (ai2sa5678) 64, (biesas678)64,_he ; 


(11 735.2846-, @1gd62735-t 


[ (ar2sa5678)64, (D12845678)64,_h6, 
[ (di2s45678) 64, (bi2345678) 64y_he i 
[ (di2sas678) 64, (bi2s45678)64,_he, 


16 t 
16 t 
wt, Aig. by.t, 


16 t 
ist, diz. bio.t, 


16 t 
16 1625-37-48 - t, 


16 t, Ay3.24.t, 


16 t 
16 t 
16 t 
16 t, 1625-37-48. t, 


16 t, 15.26. b15.26. t, 
16 t, die.t, 


16 t 

16 t 

16 t 

ist, diy.t, 

t 

16 t 

16 41735-2846 - t, 


16 t, Ai2-t, 
16 t 


16 t 
16 t 


- TMPRIMITIVE SUBSTITUTION GrouUPS OF DEGREE SIXTEEN 


Order 


No. 
79, 82 


83, 86 


87 

88 

89 
90, 93 


94 
95, 98 


99 
100, 103 


104 

105 

106 
107, 110 


111 

112 

113 
114, 117 


118 

119 

120 
121, 123 
124, 127 


128, 131 


[ (ar2845678)64,(Dr2sase7s)64, he, 16 t, 1735-2806. t, 
y3.24. Dig.24.t, Aiaog. Dig.24.t 


[ (ai2s45678)64,(Diessse78)64,_hs, 


(115.26-37-48+¢, 17362845 -¢ 


[ (Gi2s45678)64,(b12345678)64,_he, 
[ (aies45678)64, (Di2s4s678)64, he, 


[ (a12345678) 64, (b12345678)64, he, 


[ (ai2sss678)64,(Di2345678) 644 16, 


1y5.26-37-48-1, G12. di2.t 


[ (ar2s4s678)64,(Di2345678)64,_h6, 
[ (a12845678) 64, (D12845678)645_his, 


(115.26-37-48-t, diz. bi2.t 


L (12945678) 64, (D12345678)64,_h6, 


[ (ar2845678) 64, (Dr2345678)64,_h6, 


14324 - t, 114.23-56- t 


[(Gies45678)04,(D12945678)64, he, 
[ (ai2345678)64,(D12345678) 64, he, 
L (12345678) 644 (bi2345678) 645 hs , 
[ (ai2315678)e4,(O12345678) 64, his, 


Qy324.t, 4.23.56.¢ 


[(i2s4s678)64,(Dios4se78) 64, he 
[ (ar2845678) 64, (Di2ss5678) 644 hs, 
[ (ai2s45678)64,(Di2s45678)64, he, 
[ (ai2s4s678)64, (D12345678)64,_ho, 


y625-37-48-¢, 1735-2846 -¢ 


[ (ai2945678) 64, (Di2s4s678)64, ho, 
[ (ai2345678)64, (D12s45678) 64, he, 
[ (ai2sss678) 64, (Di2s4s678) 64, he, 
[ (ar2345678) 64, (Di2s4s678) 64, he, 


Ay... by.t, dsg.bi2.t 


[(ai2s15078)64, (Di2s45678)04, hs, 


yo. by. t, sg. Dy. t 
[ (a12345678)128(Diesis67s)128 18, 


143-24. f Qy423-56.¢ 


16 t, 1324+ t, 


16 t 
16 t 
16 t 
16 t, Ay2.56-t, 


16 t 
16 t, Qy2.56.t, 


16 t 
16 t, Ay2.56-t, 


16 t 
16 t 
16 t 
16 t, rz.56.t, 


16 t 
16 t 
16 t 
16 t, Aiso4.t, 


16 t 
16 t 
16 
16 1625-37-48 +E, 


16 t, 3625-37-48. t, 


gt, Aye.56.t, 


14 


Order 


No. 
132, 135 


136, 139 
140, 143 
144, 147 
148, 151 
152 
153 
154 
155, 158 
159, 162 
163, 166 
167, 170 
171, 174 
175, 178 
179 
180 
181 
182, 185 
186, 189 


190, 193 


Grace Marie BAREIS 


[ (aiegase78) 128 (bi2345678) 128. I8 ; 


(113.24.4, Ay423.56.¢ 


[ (aie345678) 128 (Di2s4s678) 128.8 , 


(113.24.4, Ay423.56.¢ 


[ (dr2345678) 128 (Di2s45678)128 |8 , 


(113.24.t, A1493.56.¢ 


[ (aiesas678) 128 (bros45678)128_]8 ; 


(13.24.¢, A1423.56.¢ 


[ (d12345678) 128 (Dios4s678) 128 |g ; 


(113.24. , Ay423.56.¢ 


[ (a12845678)128 (Dies45678) 128 |e, 
[ (12345678) 128 (Di2s45678) ves |e, 
[ (a12345678)128 (D12345678) 128 Is, 
[ (12345678) 128 (bi2345678) 128. ]s : 


Qy2.t, 1324-57-68. ¢ 


[ (ai2345678) 128 (b12345678):128 Is, gt, 


Qyg.t, A1324.57-68.¢ 


[ (aizsase78)128 (b12345678) 128 |e, 8 t, 


13.24.57-68 +6 


[ (ai2g45678) 128 (biosase78)128_|8 ; 


Ayo.t, 1324-57-63 -¢ 


[ (12345678) 128 (D12345678)128 Is, 8 t, 


Qy2.t, M1324.57-68-¢ 


[ (d12845678) 128 (Die345678) 128 Is, 


a2. on 1324-57-68 « t 


[ (ai2345678)128 (Di2sas678) 128 Is, 
[ (12345678) 128 (bi2s45678)128_|s : 
[ (ar2845678)128 (Di2345678) 128 Is, 
[ (ai2345678)128(Diesase78)128 Is, 8 f, 


y304.t, 14.23.56. ¢ 


[ (Giogase78) 128 (D12345678)198 |e, 8 t, 


y324.¢, Ay4.03.56-¢ 


[ (dizzase78)128 (D12345678) 128 |p, 8 t, 


11324. t, 114.23-56 - t 


st, 
3 t, 
st, 
st, 
st, 
gt 
gt 


gt 


3 t, 


3 t, 


3 t, 


gt 
gt 
gt 


112.56 « f, 


142.56 + f, 


2.56 « t, 


142.56 + t, 


12-56 t, 


13.24.57-68 +t, 


43.24.57-68 + t, 


Qy2.t, 1g04.57-68-t, 


13.24.57-68 +t, 


43.24-57-68-t, 


13.24.57-68 +L, 


42-56 + t, 
42-56 + t, 


42-56 « i 


IMPRIMITIVE SUBSTITUTION GrouUPS OF DEGREE SIXTEEN 


Order 


1536 


1152 


1024 


No. 


194, 197 [(arosase78)128(Di2s45678)128 Is, 8 t, Aie-56-t, 


324-4, 14.23-56-¢ 


198, 201 [[(aresis67s)128(Diesase7s)128 Js, 8 t, Aie-s6-t, 


14324 - t, (114.23-56- t 


202, 205 [ (aieg4s678)128(Diosase7s)128 18, 8 t, Aie-s6-t, 


206 
207 


- “ 


AW SHS Pb 
OPM HON 


“ 


—_ 
_ 


_ - “ 


- 


Qy304. t, 14.93.56. ¢ 
[ (aiegase78) 128 (D12345678)128 Js, 8 t 


[ (ai2845678) 128 (Di2sss678)128 |s, 8 t 

L (aiesss67s)128(Diesis678)128 |e, 8 ¢ 

[ (ai2sts078)384(D12s45678)384_b, 2 t 

[ (ar2845678) 96, (012345678) 96, 1s, 8 t, G13-24- Dis.24.¢ 

[ (a12845678) 96,(012345678) 96, 18, 8 t, isos Diss. t 

[(ar2sss678)1925(Dizsas678)192, 1s, 4 #, G15-26-37-48-¢ 

[ (ai2ses678)192,(Dosss678)192,4, 4 t, A15.26-37-48-¢ 

[ (a12345678) 96, (012345678) 96,18, 8 t, 12.56. Dio.56.t 

[ (ai2s45078)576,(Di2sase78) 576, h, 1#, di2-die.t 

[ (ar2s45678)576, (Di2s45678)576,_h, 14, die. io.t 

[ (a12345678)48,,(Dr2345678)48,, he, 12 t, Gres. t, 
5.26-37-48-¢, A16.25-37-48-¢ 

[ (a2845678)48,,(012345678)48,, he, 12 t 

[ (a12345678)48,, (012345678) 48,4 h2, 12 t 

[ (ar2sas678)48,, (012345678) 48,5 h2, 12 t 

[ (ar2345678)24, (Di2s4s678)24,_] ¢ 

[ (ar2s45678)24,, (Di2s4s678)24,,_] # 

[ (a2845678) 2445 (Dr2845678)24,0_] ¢ 

[ (a12845678)32,5(D12845678)321_he, 16 t, 15-26-37-48-¢ 

[ (a12845678)32,5(D12845678) 32,9 hs, 16 t, Aise4.t 

[ (a@12345678)32,(D12845678)324o_h6, 16 t, Arses.t 

[ (Giossse78)s2,,(Dr2945678)32,, he, 16 £, 15-26-37-48-¢ 


[.(a2s4s678)32,, (12845678) 32, he, 16 t, is.24.t 


[ (ie45678)32,, (D12846678)32,, he, 16 t, Ais.o4.t 
[ (ai2945678)32,(b12845678)s2,, hs, 16 t, G15.26-37-48-¢ 
[ (Gr2s45678)32,, (Di2sis078)32,, hs, 16 t, G12-56.¢ 
[ (Gi2sss678)32,,(O12845678)32, hs, 16 t, Ai2.s6. 


15 


GracE Marre BAREIS 


Order No. 


19, 20 [ (a12345678) 324, (bi2345678)32;,_h6, 16 t, 1735-2846 -¢ 
21, 22 — [(ai2s45678)32,, (b12345678)32,, h6,.16 t, A1.56-t 
23, 24 [ (a12345678)32,, (O12345678)32,, he, 16 t, 1625-37-48.¢ 
25, 26 — [_(aizsase78)32,,(Di2345678)32,, he, 16 t A13-24-57-68-¢ 
27,28 — [(ar2sas678)s2,,(O12345678)32,, ts, 16 4, Ar7se2sa5.t 
29, 30 [ (12345678) 321, (D12345678) 321, 116, 16 t, Ay2.56-¢ 
31, 32 [ (aies4s678)32,, (D12345678)32,,_hs, 16 t, Ay2.5768-¢ 
33, 34 [(ais45678)32,, (012345678) 3247 he, 16 t, 15-26-37-48. 
35, 36 —_[(aies45678)32,, (b12345678)321, he, 16 f, A15-26-37-48-¢ 
37, 38 — [(aie45678)324_(O12945678) 3244 his, 16 t, 1526-37-48 +t 
39,40 — [_(airsas678)324,(D12345678) 3214 116, 16 t, A13-24-56-E 
41, 42 [ (12345678) 324 (012345678) 3215 hi6, 16 t, 3.24.56.¢ 
43,44 — [ (ar2s45678)321,(012345678)321, 6, 16 t, A12-34.¢ 
45, 46 [ (G12345678)3215 (012945678) 32,4 he, 16 t, Qyo.34.¢ 
47, 48 [ (a12845678) 64, (D12845678) 645 Is, gt, M13.24-57-68-¢ 
49, 50 L (a12345678) 64, (Di2s45678) 645 18, gt, Gy3.24-57-68-¢ 
51,58 — [(qiesss67s) 64, (Di2s5678) 64, 1s, 8 t, G1735-2806-t, 
Qy2.56.t, Q17-28-35-46-t, 1493.56 - t, Ay. bi. t, 
Ayg04. Dy2.t, Aig.24.t 
59, 60 — [(aias45678) 64, (Di2s4s678) 64, 1s, 8 ty Aia-56.t 
61,64 — [(diess5678) 64, (Di2sas678)64, 18, 8 t, G1735-2846-t, 
ay... by2.t, 1526-37-48. ¢ 
65, 66 —_ [_(diesis678) 64, (Dizsase7s)64,_18, 8 t, A13-24.t 
67, 68 [ (a2345678) 64, (D12845678)64,_ls, gt, Qy403.56-b 
69, 70 — [(aiesss678)64, (Dizsas678)64, 1s, 8 t, Arzs5.2846- 
71, 72 — [(aiesss67s)64, (Di2s45678)64, 18, 8 t, G1735-2846.t 
73, 74 [ (ai2s45678) 64, (Di2s45678) 645 Is, gt, A1735-2846.¢ 
75, 76 — [(diesss678)64, (Dizs4s678)64, 18, 8 ty Ais.24-¢ 
77, 80 — [(aizs45678) 64, (Dizs4s678)64, 1s, 8 t, Gi2-56-, 
yg. bys. t, As6. O12.t : 
81, 82 — [ (aizss5678)64, (D12845678)64, 18, 8 t, Qi2-56-t 
83, 86 — [(ai2s4s678)64, (Di2sis678)645 18, 8 t Gi2-56-t, 
Ayo. bist, sg. D1o.t 
87, 88 — [(ar2s45678) 64, (Di2sis678)64, 18, 8 t, Ai2-56.¢ 


= > 


ls meee Mail tae | ani iil 


ee ee ae ae ee 


“ie 
” . 


IMPRIMITIVE SUBSTITUTION GrRoUPS OF DEGREE SIXTEEN 


Order 


No. 
89, 92 


93, 94 
95, 98 


99, 100 
101, 110 
111, 115 
116, 119 
120, 121 
122, 123 
124, 127 
128, 129 
130, 137 
138, 141 
142, 145 
146, 149 


150, 153 


154, 155 
156, 159 


160, 161 


[ (ar2sss078)64,(Dresis67s)o4, 1s, 8 t, Are-s4-t, 
Ay. by. t, 34. bya. t 

[ (a12345678) 64, (Di2ssse7s)o4, 1s, st, Cie.s4.t 

[ (ar2sss678)64,(Di2si5678)64, 1s, 8 t, Ar2-a4.t, 
Ayo. bya. t, Aigz4. Dy. t 

[ (aressse7s) 64, (Drosise7s)o1, 1s, 2 t, i234. 

[ (a2s4s678) 61, (Drosaso7s)o4, Is, st, is-o4-t, Gi2.s6.t, 
324.56. t, 4625-37-48. t, 43.24.56. O12. t, yg. by. t, 
Ase. Dy. t, 17362845. t, 3304. Oyo. t 

[ (12345678) 64, (Di2sase7s)64, Is, 8 t, Ais-24.t, 
1625-37-48-, Gyo. Di2.t, Ar736-2845. Dio. t 

[ (aizsase7s)64,(Dizais678) 64, 18, st, A12-56-t, 
Aya. by. t, Ase. bia. t 

[ (ai2s45678) 64, (Di2s1s678)64, Is, 8 t, A13-24-57-68-¢ 

[ (ai2s45678) 64, (Dios4s678)04, Is, st, A1s-24-57-68-¢ 

[ (a12845678)64,(Diosase7s)o4, Is, 8 t, 12. dis. t, 
Ay2.56-t, dsg. bia. t 

[ (a12845678) 64, (D12845678) 64, 1s, 8 t, Ar2-56-t 

[ (a12845678)64, (12845678) 64, 1s, st, i2-t, dis.24.57-68-, 
493.57-63+t, 13-24. D13.04.t, Aiso3- Diz.24.1, 
112.57-68+ Dis.04 F z; 57.68 + Di3.04 .t 

[ (12345678) 64, (Dr2sas67s)e4, 18, st, A13.24-57-68-E, 
(113.24. Di3.04 ‘ t, 157-68 + Dy3.04 t 

[ (a12345678) 64, (D12s45678) 64,18, 8 ty, diet, 
Ay423.57-68-t, Aise7.35-46.-¢ 

[ (ai2315678) 64, (Di2sase78) 61, Is, st, A15-26-37-48-¢, 
43.24. Di3.24 * ‘ (116482537 - Diszo4 »t 

[ (ai2845678) 64, (D12845678)64, 1s, 8 ty Ar2-56-t, 
(143.24. Dig.24 ° t, 423-56 + Diz.04 at 

[ (arzs45678)64,(Di2sss678) 04,18, 8 t, Ai2-56-t 

[ (ai2s45678) 64, (Drosase7s) 1, 1s, st, Ai2-s6-, 
(143-24- Dyz.24 e t, 4423-56 + Dy3.04 Re 


L (aizsss678) 64, (Di2s4s678) 64, Js, gt, di2.56.t 


17 


18 


Order 


No. 
162, 171 


172, 175 


176, 183 


184, 185 
186, 187 
188, 191 


192, 193 
194, 195 
196, 197 
198, 201 


202, 203 
204, 211 


212, 215 
216, 219 


220, 221 
222, 223 
224, 225 
226, 227 
228, 229 
230, 231 
232, 233 
234, 235 


GRACE MARIE BAREIS 


[ (aresas678) 64, (D12s4s678) 64,_Is, gt, dy2.t, M13.24.57-68-t, 
(y493-57-68-¢, C1735-2846-t, C1gs62735-t, 


13.24. D13.24-¢, A143. D1g-24.t, 57-68. Dis-24.F, 


(12.57.68 - O13.24.¢ 


[ (ar2sase78)64, (Des45678)64, 8, gt, dy.t, 


1735-2846-t, 18462735 ¢ 


[ (a12345678)64, (D12sas67s)64, 8, 8 t, diet, 
13.24-57-68-t, Q1423-57-68-t, 13-24- dig.24.t, 


11493. by3.24.t, 57-68-b13-24.t, 12-57-68 D1g.24-F 
[ (atosas678)64,(Dizsas678)64, 1s, 8 t, Aie-56-t 


[ (ai2345678) 64, (Di2s45678)64,_18 
[ (a1o345678) 64, (Di2s45678)64,_18 


st, 
,8t, 


013.24. D13.04.t, Gr4o3- Dig.24.t 


[ (ai234s678) 64, (Di2345678)64,_13 
[ (12345678) 64, (bi2345678) 64518 


[ (2345678) 64, (b12345678) 64 Is ; 
[ (cu2sas678) 645 (012345678) 645_18 ; 


(56. Dae oH 


[ (a2s45678) 64, (Dr2sasers)e4, Is, 


sf, 
st, 
3 t, 
3 t, 


3 t, 


142-56 + t 
142.56 - 


142.56- t 

(143.24.57-68 +6 
(43.24-57-68 +t 

142.56 + t, a2. Dye Fy t, 


(142.56 t 


[ (ai23a5678)64,(Di2345678) 64,18, 8 t, Gi2-s6-t, Arse. t, 


14.03-56-t, A15-26.37-48-¢, Ar7362845-t, dia. die. t, 


(13.24. Dip. t 


[ (a12345678) 64, (Di2s4s678) 6418, 8 t, dio-se.t, 


ay. Dio. & 56. bio.t 


: L (12345078) 64, (b12845678) 64,_18 


st, 


G2. bya. t, 1736-2845. Die. t 


[ (2345678) 64, (bisas678) 64,_]s 
[ (ai2345678) 64, (b12345678) 64, _18 
[ (aios45678) 64, (D12s45678) 645_Is 
[ (aies45678) 645 (b12345678) 64y_|s 
[ (aizg45678) 64, (D12845678) 64,_Is 
| (di2345678) 64, (bizsas678) 64, Is 
[ (a12345678) 64, (b12845678) 64,_|s 
[ (a12345678) 64, (Droa45678)644_]s 


8, 
st, 
,8t, 


, at, 


,8f, 
,t, 
.8t, 
3, 


15-26-3748, 


142.56 + t 
142.56 + t 
115.26-37-48 « t 
(115.26-37-48 +t 
aygo4.t 
(14.4.23-56 « t 
A324. t 
144.23-56 + t 


IMPRIMITIVE SUBSTITUTION GrouUPS OF DEGREE SIXTEEN 


Order 


No. 


236, 237 
238, 239 
240, 241 
242, 243 
244, 245 
246, 247 
248, 255 


256, 263 


264, 267 
268, 271 


272, 273 
274, 277 


278, 281 
282, 285 
286, 289 
290, 293 
294, 297 
298, 301 
302, 305 


306 


[ (ai2345678) 64, (Di2ssse7s)64,_18, 8 ¢, 
[ (ar2s15678)64, (Di2sase7s)64,_I8, 8 t, 
[ (aiasis078)64, (Di2s45678) 64,18, 8 t, 
[ (arasss078)64,(Di2s45678) 04,18, 8 t, 
[ (ai2s4s678) 64, (Di2sase78)64, Is, 8 t, 
[ (arasise7s)64, (Di2sase7s)64, Is, 8 t, 
[ (ai2s4s078)64, (Dizssse78)04, Is, 8 t, 


1925.37-48-t, Gi735-2816-t, Gie.di2.t, ase. die.t, 


113.24. by ° i, 1423-56 + bye .t 


[ (dress5678)64, (Di2sase7s)64, Is, 8 t, 


Ay625-37-48-t, Ar735.2846-t, Gio. die.t, dg. dist, 


(143.24. bie : t, (11423-56- bio & 

[ (ar2s4s678) 64, (Dr2sase78) 04, Is, 8 t, 
a2. b.t, (17352846. ¢ 

[ (aresss67s)64, (Di2sase7s)o4, Is, 8 t, 
Ay. bye. t, sg. Dy. t 

[ (ai2s45678) 64, (Diessse7s)e4, Js, 8 t, 

[ (@i2s45678)128(Diesas67s)128 14, 4 t, 
13.24-57-68-t, 1423-5867 - t 

[ (Gr2sss678)128(Dr2sss678) 1284, 4 t, 
(113.24-57-68-t, 1493-5867 « t 

[ (aiossse7s)128(Dr2s4s078) 1284, 4 t, 
13.24.57-63-1, 1423-5367. ¢ 

[ (ai2945678) 128 (Diesss678)128 hk, 4 f, 
(y3.024-57-68-t, 1423-5867 - ¢ 

[ (a2845678) 128 (Dizsis678) 128k, 4 t, 
3.24-57-68-t, 1423-5367 +t 

[ (12345678) 128 (Di2sis678)128 14, 4 t, 
13.24-57-68+4, 1423-5867 -¢ 

L (aies4s67s)128(Diesss67s)128 lk, 4 t, 
(13.24-57-68-4, 1423-5867 - ¢ 

[ (ai2s45078)128(Drasas678) 1284, 4 t, 


Q43.24-57-68-¢, 1423-5967 -¢ 


[ (ai2s4s678)128 (Diesss678)128 4, 4 t 


(113.24-57-68 + t 
13.24.57-68 +t 
(112.56 + t 
142.56 + t 
13.24.57-68 - 
(143.24-57-68 +t 
1324.1, 


304. t, 


11625.37-48- t, 
2-56 + t, 


Ay2.56.¢ 
42-56 + a 


119.56» i 
42.-56 « t, 
2.56. t, 


142.56 « t, 


19 


20 


Order 


896 


768 


672 
576 


512 


Grace Marre Bares 


[ (ai2s4s678) 128 (Di2gas678)128 4, 4 t 

[ (G12345678)128(Diesa5678)128 |4, 4 f 

[ (Gr2sa5678)128(Diosase7s)128 4, 4 t 

[ (aie345678)128 (Diesas67s)128 [4, 4 ¢ 

[ (a12345678)128(Dies45678)128 4, 4 ¢ 

[ (a12345678) 56(Di2s45678) 561s, 8 t, Arosase7-t, 
@1357246-¢, Gya73625-t, G1526374-, Cr642753-F, 
11765432 t 

[ (a12345678) 48,4 (Di2345678)48,4 Is, 8 t 


: [ (di2sa5678)48,,(Di2sase78)48,, Is, 8 t 


[ (12345678) 96, (012345678) 964 l4, 4 t, G15-26-37-48-f, 
A12.56+ D19.56-t, 16-25-37-48- D12.56.¢ 

[ (aes4s678)192, (Di2345678)192, 2, 2 t, Aig.c4- Dig.04.t 

[ (i2s45678)192, (Dizsase78)192, ot, dy.bi.t 

[ (aresase78)192, (Dizs45678)192, 2, 2 t, ie. di.t 

[ (G12345678) 384 (Disase78)384_h, 1¢, Gi2-s4.56-78.t 

[ (a12345678) 336 (Diesas678)336_h, 1 ¢ 

[ (a12345678) 24, (Dizsas67s)04, he, 12 t, @15-26-37-48-¢ 

[ (dies45678)24,, (Di2s45678)241, he, 12 t, A16.25-37-48-¢ 

[ (@12845678)16,, (012345678) 1697_] t 

[_(a12345678)16,, (012845678) 165_| ¢ 

[ (€12345678)1655 (012345678) 1695] t 

[ (112345678) 165 (012845678) 16y5_| t 

[ (12345678) 16, (012845678) 1650_| t 

[ (a12345678) 1695 (012345678) 1699_| 

[ (a12345678)32,(012845678)3249.18, 8 t, Aisoa.t, 
115-26-37-48-, rzaeosd5-t 


| [ (Gi2ga5678)32 49 (012845678)3249_18, 8 t 


[ (di2345678) 321 (012845678) 32418, 8 t 

[ (a12345678)324 (012845678) 3241s, 8 t 

[ (qi12345678) 32, (012345678) 32,518, st, Arsea.t, 
142.56 - Diz.56 = t, 1134.56 » Dyo.56 Py 

[ (12345678) 321 (012845678) 321918, 8 t, Aigo. t, 
142.56 + Dyp.56- - 134-56 - Dy2.56.t 


Order No. 
22, 25 . 


26 
27 
28 

29, 32 


a = a ee ee ee ee ee ie eee Tee 
i e hing = ‘ . 


33, 36 


<a 37 
= 38, 41 


42, 47 


> 
& 
~I 
eo 


IMPRIMITIVE SUBSTITUTION GROUPS OF DEGREE SIXTEEN 


[ (Giessse7s)32,,(Or2345678)32,, 18, 8 t, Ars.24.F, 
A15.26-37-48-t, G1735-2816-¢ 

[ (aiesase7s)s2,,(Dresase78)32,, 18, 8 t 

[ (arosss678)32,, (Di2sis678)32,, 18, 8 f 

[ (ai2s45678)32,,(D12845678)32,, 18, 8 t 

[ (Gresss67s)32,,(Or2s45678)32,, 8, 8 t, iesa-t, 
43.24. t, Aj4.23.¢ 

[ (Giosss678)32,,(O12345078)32,, 18, 8 t, Ar2-s4-t, 
3.24.1, (y2.56- Dig.56.t 

[ (ai2g45678)32,, (b12345678)32,,_I8. 8 t 

[ (ar2s4s678)32;» (Di2sis678)32,. 18, 8 t, 15-26- b15.26.t, 
13.24-57-68+¢, Ai753-2846.¢ 

[ (a12845678)32,»(Di245678)321, 18, 8 t, ie. Die. t, 
Ay2.56-t, 56. Di2.t, G15.26-37-48-t, 1625-37-48 - Die. t 

[ (Gs2s45678)32,4(012815678)32,, 18, 8 t, Gi2-s6-t, 
145.26437-48+t, G16.25-37-48-¢ 

[ (ai2s4s678)32,(D12945078)32,, 1s, 8 t, Aio-s4-t, 
1z5.46 - 035-46 « t, 19-3645 - 035-46. t 

[ (Grz245678) 32,,(D12345678)32,, 18, 8 t, A13-24-57-68-¢, 
Ay2.56-t, 324-5768 - t 

[ (@12545678)32,_ (b12845678)3215 Is, gt 

[ (a12345678)32,,(D12345678)32,, Is, 8 t 

[ (a12345678)32,, (Dr2845678) 32,5 18, 8 ¢ 

[ (@i2915678)32,(012845678)325 18, 8 t, ia-s4-t, Gs4.s6-t, 
2-35-46 « bi2.35-46 t, (13645 + Dyo.36-46 « t 

[ (ar2ses678)32, ,(b12345678)3215 Is, gt 

[ (aresss078)32,,(D12845678)32,, 18, 8 t, Gi2-56-t, 
45.26-37-48-1, ayo. by. t, Gygae-37-48- Dig. t ? 


[ (ar2345078)32,,(D12845678)32,, 13, 8 t 


[(qu2sss678)32,4(b12345678)32,, 18, 8 t, Ai2.s4-4, 


Ay2.56.t, y3.24. 013.04. t, 1493-56 - 013.04. t 
[ (ar2s46078)32,(Pr2946078)32,, 1s, 8 ¢ 
[ (ai2sss678)32,,(Pr2346678)32,, 18, 8 t, Gie-a4-t, 
y2.35-46 - 012.35-46 +t 


21 


22 


Order * 


No. 
84, 87 


88, 91 


92 
93, 96 


97, 101 
102, 104 


105 
106, 108 


109, 113 


114, 118 


119, 123 


124 
125, 128 


129, 1382 


133 

134 

135 
136, 139 


140, 143 


144 
145 


Grace Marie BAREIS 


[ (a12845678)32,, (12345678) 324,18, 8 £, 1735-2846. t, 
113.24 by3.04 . K 57-68 » Di3.04 oe 

[ (a12845678)32,5 (Dizsase78)32,, 18, 8 , 
A1735-2846-t, G2. by2.t 

[ (a12845678)32,,, (D12845678)32,,_18, 8 t 

[ (12345678) 3214 (bi2345678)32,5_l8, st, 
Ay. by2.t, 1324-57-63. Dio. t 


12.56 + t 


(11735-2846 - t 

[ (dizs45678)32,, (Di2s4s678)32,, 18, 8 ty io-s6-t, 
1625-37-48 t, ayo. bio ° a 56. Dis 5 

[ (ai2345678)32,, (D12s45678)32,, 18, 8 f, A13-24-57-68-t, 
1625-37-48 t 

[ (412845678) 32,, (Di2sase78)32,, 18, 8 ¢ 

[ (d12845678)32,, (D12845678)32,, 18, 8 f, 
1625-37-48 +t 

[ (a12345678)32,,(D12845678)32,, Is, 8 t, 
13.24. b13.04.¢, 13.24.57-68-¢, Ar7362845.¢ 

L (a12845678)32,, (Di2345678)32,, 18, 8 t, Aia-56-t, 
17362845 + a ay. Dis ‘ t, 156 - Dye ae 


(113.24.57-68+l, 


57.68 + Dy3.24. t; 


[ (d12345678) 32,5 (D12845678)32,, 18, 8 t, Ai2.s6-t, 
17362845 - t, dh. by. t, sg. Dig. t 

[ (ar2845678)32,, (D12345678)32,, 18, 8 f 

[ (a12845678)32,, (D12815678)32,, 18, 8 t, 12-5768. t, 
13.24. D13.24.¢, 14.23 - Dig.24.t 

[ (ai2315678)321,(Di2s45678)321, 18, 8 ty A15-26-37-48- 6, 
(y2.5768+t, A1736-2845.t 

[ (a12345678)32,, (D12s45678) 32,7 18, 8 t 

[ (d12345678)32,, (012845678) 324718, 8 t 

[(@12845678) 3217 (Di245678)3217_1s, 8 t 

[ (12345678) 321, (D1245678) 324, 18, 8 t, Are-5768-t, 
13.24. 13.04.t, 1537-2648 - D1g.04.t 

[ (d12345678)32,, (012845678) 321,18, 8 t, G13-24-56-t, 
15.26-37-48-¢, Ar7s5osa6.t 

[ (12345678) 3215 (012845678) 3215 1s, 8 t 

[ (a12245678) 321, (D12845678)324 18, 8 ¢ 


IMPRIMITIVE SUBSTITUTION GROUPS OF DEGREE SIXTEEN 


Order 


No. 


146 
147, 150 


151 

152 

153 
154, 157 


158 
159, 162 


163 
164, 167 


168 
169 
170 
171, 178 
179, 182 
183 
184, 191 
192 
193, 196 
197, 200 
201, 204 


205, 208 


[ (@izs45678)32,, (D12815678) 32 ls, 
[ (dizsase7s) 32 1g (012845678) 32 es. 


45.26-37-48-, 1625-3847 -¢ 


[ (ai2345678) 32, (b12845078)3214_Is. 
[ (ai2845678) 321, (b12345678) 32 maa 
[ (@12345678)321, (Di2s45678)32 8, 
[ (a12845678) 32, (bi2s4s678)321¢_I8. 


43-24-56 + t, (11324. Diso4 % 


[ (a12845678)32,q (012345678) 325 18, 
[ (au2ss5678)32,,(012845678)326_18, 


(43.24-56 + t, (11324. Dy304 t 


[ (ai2345678) 3214 (012845678) 3214_18, 
[ (a12815678)32,, (012945678) 321418, 


13-24-56 + t, 114.2356 + t 


[ (di2345678)321, (D12315678) 32 8 : 
[ (au2345678)3219 (D12845678)32,5_18, 


[-(a12845678)32 4 (D12845678) 3216 18, 


gt 
gt, dye.34.t, 


gt 


[ (aiess5678) 64, (D12345678) 645], at, Qhye.56.t, 


143.24-57-68-E, 1423-5867 « t, Ayo. bys. t, Ase. Diz. t, 


y304.57-68- Dio. t, Ai4.23-5867 - Dio. t 


[ (ai2sss678)64, (Dizaise7s)64, hh, 4 t, Ai2-56-t, 


43.24-57-68-t, 1423-5867 - ¢ 


[(diessso7s) 64, (Drosise7)64, 14, 4 


t 


[ (arzss5678)64, (Dr2ais67s)64,h, 4¢, di2-56-t, 


Gy3.24-57-68-t, Gigos-5967-t, Gio. Dio.t, ase. die.t, 


y324.57-68-Di2.t, 14.08.5967 - Dio. t 


[ (ar2sss07s)64, (Di2945678) 64,14, 4 


t 


[ (aiesss67s)64, (Diosase7s)64, 14, at, dy.34.F, 


45.26-37-48-¢, 1625-3847 -¢ 


[ (aiesss678)64, (Di2s4s678)64,_], at, dyo.34.t, 


(145.26-37-48-, G1625-3847-¢ 


[(aross5678)04, (Dizssse7s) 64,4, 4, Gie-s4-t, 


15.26-37-48-4, 1625-3847 -¢ 


[ (aizsss678)04,(Dr2345678) 64, hh, «t, diz-s4-t, 


145.26.37-48+t, 1625-3847 - t 


23 


24 


Order 


No. 
209, 212 


213, 216 


217 

218 

219 
220, 227 


228, 235 


236 
237, 244 


245, 248 
249, 252 
253, 254 
255 
256, 263 


264, 271 


272 
273, 280 


281, 284 


GrRacE Marie BAREIS 


[ (ar2s45678) 64; (Diasase7s)e4, lb, 4 t, Aio-a4.t, 
(115.26-37-48-t, Q1625-3847.t 

[ (a12345678)64, (Di2sase7s)e4, 4, 4 t, Aros, 
15.26-37-48-t, Cr625-3847-t 

[ (a12345678) 645 (Dizsase7s)64, 14, 4 ¢ 

[ (ai2345678)64, (Dizss5678)64,_, at 

L (d12345678) 64, (Di2s4s678)64, 14, 4 t 

[ (a12845678) 64, (Di2sis678)64, 4, 4 t, A13-24-57-68-E, 
Gyo.56-t, Aiges.5768-t, Ai. bdy2.t, Aigos-57-68- Dit, 
sg. Dip.t, Aiz.24-5768- 012-6 

[ (12345678) 64, (Diesase7s)e4, 4, 4 ¢, 13.24-57-68-E, . 
Q12.56-t, Aigos-5768-t, Aig. Dio.t, Aigos.57-68- Dia. t, 
sg. bi2.t, 13.24.5768. Dio. t 

[ (12345678) 64, (Di2sas678) 64, la, 4 t 

[ (diesas678)64,(Diosase7s)e4, 4, 4 ¢, Gie-s6-t, die.t, 
5g.t, 33.24. D1g.04.t, Aiso3.56-D13.24.4, Gr42g. D13.04.t, 
33.24.56 - D13.24.t 

[ (ai2s15678)64,(Di2s4s678)64, 4, 4t, Giz.s6-t, diet, As6-t 

[ (dizss5678) 64, (Diosase7s)e4, 4, 4 t, Aio-se-t, Gio.t, ase.t 

[ (a12345678) 64, (Di2345678) 64, 14, at, 13.04. b33.04.¢ 

[ (a2345678) 64, (Dizsas678)64, 4, 4 t 

[ (Giesas678)64, (Diesase7s)e4, 4, 4t, Ar2-56-t, 
13.24.57-68-¢, C1423-5967-¢, A1g.24- Dig.24., 
1493.56 + Dig.04.t, A57-68-13.04.¢, A12.5967- Di3.24.t 

[ (i2s45078) 04, (Diosase7s)e4, 4, 4, Aie-s6-t, 
13.24-57-68-t, G1423-5967-t, 13-24. D13.24., 
1493.56 D13.24.t, 57-63. ig.04.t, 12-5867 - b13.04.t 

[ (ai2345678)64, (Di2s4s678) 64, le, 4 t 

L (12345678) 64,(D2sas678) 64, lk, 4 t, Ar423-5867-F, 
Ay2.56-t, 14.23.58-67-t, Cia. dy2.t, Ai4.03-5867- Dio. t, 
ase. by. t, 1493-58-67+ Dio. t 

[ (Gres45678)64,(Diosase7s)e4,l4, 4 t, G1423-5867- t, 
2.56, A14.23.58-67-¢ 


IvpriMitivE SusstiruTion Groups or DEGREE SIXTEEN 25 


~." Oxder No. 


— 285, 292 [ (arsassezs)e,(Dissise7s)er, kk, 4 t, Crs2s.sse7-t, 
Gy2.56.t, G14.23-58-67-t, Qie.Dio.t, di4.23.5867- Dia. t, 
Ag Dy. t, 1423-58-67 Dia. t 
293 — [ (Gizs4s678)64,(Diosiso7s)o4, h, 4 t 
294° [ (aiases678)64,(Dr2esise7s)o4, 4, 4 t 
295, 298 — [ (aiessse7s)64,(Diesise7s)64,_l4, 4 t, Aio-s4-t, 
16-25-37-48-, 1526-3847 -¢ 
299, 302 [(aressse7s)04,(Drosase7s)e4, 4, 4 t, Ar2-s4-t, 
116.25-37-48-t, C1526-3847-¢ 
303, 306 — [(ar2sss678)64,(Diosiso7s)64, hk, 4 t, Aie-s4-t, 
y6.25.37-48-t, 1526-3847 -t 
307, 310 — [(ar2545678)644(Di2345678) 64,4, 4 t, Ci2.s4-t, 
ae 6.25-37-48-t, 1526-3847 
311, 314 — [(ar2s15678)64,(Di2sis078)64,h, 4 t, die-s4.t, 
y6.25-37-48-t, 1526-3847-t 
315, 318 — [(ai2s45678)64,(Di2sis678)61, 4, 4 t, Giz.s4-t, 
116.25-37-48-, Aise6.3847-¢ 
319 [ (Gres4s678)64,(Dr2345678)64, 4, 4 t 
320 [ (ai2ses678)64,(Dr2sis678)64,_l4, 4 t 
321 (auzssse7s)64,(Diossse7s)e4, ls, 4 ¢ 
322, 329 — [(aiosase7s)o1,(biosiso7s)o4, 4, 4 t, i2.56.t, 
13.24-57-68-t, Ais2g-5867-t, Ae. Die.t, ds6.bi.t, 
y324.57-68-t, 14-23-5867 - Dio. t 
330, 337 — [(aiessse7s)64,(bi2sss678)64,_h, 4 t, Ar2-s0-t, 
Gy3.24-57-68-t, G1423-5867-t, Gir. Die.t, ds6.dy2.t, 
G1g24-57-68-Di2.t, 14-03-5867 Dio. t 
: 338 [ (ar2sis078)04,(Di2ssse78)64, 4, 4 t 
: 339, 340 — [ (aresise7s)128(Diesise7s)128 b, 2 t, Gie-g4-t 
341, 342 — [_(aro945678)128(Di2ssse78)128_b, 2 t, Are.s4-t 
343, 344 — [_(ai2545678)128(Dizs45678)128_b, 2 t, diz.sa.t 
B45, 346 — [(ai2sss678)128(biosss678) 128 b, 2 t, Gie.s.t 
B47, 348 — [(aiessse7s)128(biosiso7s)128 b, 2 t, di2.s4-t 
349, 350 — [(ar2sis67s)128(Dizsis078)128_b, 2 t, Ai2.s4-t 
351, 352 — [_(Giessse7s)128(biessse78)128_b, 2 t, Are.s4.t 


396 


256 


: GrRAcE Marre BAREIS 


[ (Grosase78)128 (D1o345678)128 2, 2 t, Ar2.g4.¢ 

[ (12345678) 192,(D12345678) 192, h, 1 t 

[ (a12345678)192, (Di2s45678)192, h, 1 t, G12-34-56-78-¢, 
13.24+ Dig.04.¢, C14.23-56-78- Dig.o4.t 

[ (a12345678)192,(Dizsas678)192, , 1, Gi2.s4-56-78-f, 
a2. bys F t, 134.56-78 + Dye mS 

[ (aies45678) 192, (Dios45678)192, 1, 1 t, Ar2-34-56-78-¢, 
ayo. bye P i. 134.5678 » Dio 4 

L (d12345678) 192, (b12345678)1925_h, it, Q12.34.56-78-b, 
aye. bye 5 ‘ 134.56-78« bie at 

[ (2345678) 192, (Dizsas67s) 192, ht, 1t, y2.34.56-78 +f, 
42. Dye 4 t; 134.56+78 « Dre 2t 

[ (a12345678) 96, (012345678) 96.2, 2 t, Ae. dio.t, 
13.24. Dy3.24.t, C1423. D1403.t 

[ (di2345678) 4815 (012845678) 4815 4, 4t, M15.26.37-48.¢ 

[ (Gr2g15678)48;, (0123456784816, 4 t, 15.26.3748. 

[ (ai2g45678)24,5(Or2345678) 241018, gt, M135.246., 
13.24.78 - b13.04.78-¢ ; 

[ (12345678) 24, (Di2sase7s)24, Is, 8 t, Ai2-56- Die.56.¢ 

[ (a12845678) 168, (D12345678)168,_h, 1 

[ (@iz345678) 168, (D12845678)168,_h, 11, 15649378 - Dis642378  t 

[ (12345678) 16), (012845678) 169,18, 8 t, Aiz-56.t 

[ (a12345678)16,, (012345678)1657 18, 8 t, A1se4-5768-¢ 

[ (a12345678)16,, (D12345678)16)7 18, 8 t, C1256. 

[ (@12345678)1657 (012845678) 169, 1s, 8 t, 15-26-37-48-¢ 

[ (12345678) 165, (D12345678)16)7 18, 8 #, G12-56-t 

[ (4112345678) 1655 (D12345678)16,; Is, 8 t, C125-3847-t 

[ (12345678) 16), (012845678) 1695 1s, 8 t, G13-24-57-68-t 

[ (a12345678) 16,5 (012345678) 16,5 1s, 8 t, Ar2-56-t 

[ (@i2345678)16,, (D12845678) 1655 1s, 8 t, Gi2.56-t 

[ (12345678) 16,5 (O12345678)16,; 18, 8 t, Ai2-56-t 

[ (12345678) 16, (012345678) 1651s, 8 t, A1735-2846- 

[ (d12345678) 169, (012845678) 16), 18, 8 ty iz-a4-¢ 

[ (a12345678) 165 (012845678) 165518, 8 #, Grza6zs15-¢ 


Order 


No. 
27, 28 
29, 30 
31, 32 
33, 34 
35, 36 
37, 38 
39, 42 


43, 44 
45, 46 
47, 52 
53, 56 


57, 62 


63, 65 


‘66, 67 


68, 70 


71, 72 
73, 74 
75, 85 


86, 87 
88, 89 
90, 92 


93, 94 


ImMPRIMITIVE SUBSTITUTION GrouPS OF DEGREE SIXTEEN 


[ (a12945678)165, (012345678) 169518, 8 t G12.34.¢ 

[ (12345678) 16,5 (012845678) 165018, 8 t, 15-26-37-48-¢ 

[(a12345678)16,o(012345678) 16,018, 8 ty A1s.24-56- 

[(@r2s15678)16,(012345678)1650_18, 8 t, 1526-37-48 -¢ 

[ (a12545678) 16,5 (12345678) 169 1s, 8 t, G1625-3847-¢ 

[(a12845678)16,, (012845678) 1695 1s, 8 t, 13-24-56. 

[ (a12345678)32,5(012845678)3210_l4, 4 t, Aiea, 
112-56. Dio.56- t, 13456 « Dyo.56.t 

[(ai2345678)32,o(Di2s4s678) 3219, 4 t, dies. t 

[ (ar2845678)325(12345678)3210, 4 t, Grse4-5768-¢ 

[ (12345678) 32,9(O12845678) 324 14, 4 @15-26-87-48-¢, 
17362845 - t, 2.56. Di2.56-t, 34.56 D12.56-t, 
16.25-37-48- D12-56-t, 1526-3847 - Diz-56.t 

[ (12245678) 32,5(012345678)321.4, 4, Ar2-34.t, 
2.56. 012.56. t, g4.56- O19-56.t 

[ (Guosis678)32,,(Dresase7s)32,,., 4 ¢, Ara-s4-t, 
115.25-37-48-t, 1625-3847 +t, 2.56. Di2.56.t, 
4116.95-37-48 - Di2.56.¢ 


| [ (@r2g1s078)2,, (D12345678) 324-14, 4 Ay.%.t, 


42.56 - Dyo.56- ‘ 846-2735 + Dy2.56 e : 

[ (Guesase7s)32,; (Dressse7s)32,, 4, 4 t, Gr2-s4-t 

[ (Gr2s4s678)32,,(Diesise78)32,,., 4 t, Aio-s4-t, 
12.56. 019.56. t 

[ (@yo345678) 32, ,(D12345678)32-,l4, 4 t, G13-24-57-68-4 

[ (araass678)32,;(Di2sss678)2,,.4, at, G12-04-t 

[ (ar2816678)32,(D12345678) 3219-14, 4 t, i3-24-57-08-t, 
Ay2.56.t, Aises-s768-t, 15.26-37-48-t, die. by.t, 


(1423-57-68 + bye ° t, 143.24. Di3.04 < t, 11423.-56 + Dy3.24 ° t, 


Ar40g- Di4og.t, g4-57-68- Dios. t : 
[ (ai2ass678)32,,(Di2sise7s)32:.1, 4 t, G15-26-37-48-t 
[(ar2s45678)32,,(D12845678)3215h, 4 t, G13-24-57-68-¢ 
[ (Gies45678)3215(D12945678) 3251, 4 t, Arse4-5768-t, 
15.26-37-48-t 
[ (ai2s45678)32,, (D12345678) 32,44, +t, 15-26-37-48-t 


27 


107, 110 


111, 112 
113, 114 
115, 118 


119, 120 
121, 122 
123, 124 
125, 126 
127, 128 
129, 130 


131, 138 


139, 142 


143, 144 


145, 148 


149, 150 
151, 152 
153, 154 
155, 156 
157, 158 
159, 160 
161, 162 
163, 169 


Grace Marie BAREIS 


[ (a12345678) 321 (012845678) 324 4, 4 t, Gi2.56.¢ 

[ (a12345678) 3215 (D12345678) 324, 4 t, Cisea-szes-t 

[ (a12845678) 32» (D12845678) 3245-1, 4 t, Gi2-s4-t, 
y2.56-t, 15.26-37-48-4, Ci1g25-3347-¢, 16.25-37-48-t, 
13.24. Dig.04.t, 1537-2648 « D1g.04.¢ 

[ (a12345678) 32,5 (012845678) 324k, 4 t, i2.56-t, 
Gy3.24. Dig.04.t, Gre37-2518- 13.24. t 

[ (a12845678) 325» (D12845678)3239 4, 4 t, Gie.s4-t 

[ (12345678) 32,5 (D12845678) 3215 4, 4%, Gir.s4-t 

[ (a12345678)321» (012345678) 324k, 4 t, Giz.s4-t, 
13.24. Dy3.04.t, 1537-2648 +t 

[ (12345678) 3215 (b12345678) 3215, at, (15.26.37-48-¢ 

[_(a12845678) 3245 (Di284s678) 3249-1, 4 t, Ar625-3807-t 

[ (12345678) 324» (D12845678) 325, 4t, Ai2.56-¢ 

[ (12345678)321» (b12845678) 32154, at, Ay2.56.¢ 

L (aixs45678)32,1» (Dr2s45678)3239 lt, 4t, Aga.56.t 

[ (Gos45678)32 vo (D12845678) 3215 1, at, Cga.56.t 

[ (@i2345678) 324» (D12345678)32, 4, 4 t, Aro.s4.t, 
256-4, 15.26-37-48-¢, M16-25-37-48-¢, Ai526.3847-t, 
1493. Di403.t, G16492587 - Drang. t . 

[(a12845678)32,4(Di2345678)32,, 4, 44, Gi2.s6-t, 
A403. Dysog.t, 15480637 - D140 - t 

[ (G2345678)32, (012345678) 9214, 4 t, Aie.g4.t 

[ (1245678)3219(D12845678) 3294, 4 t, Ara-s4-t, 
Gy423. Di423.t, Argea- Digog.t 

[ (a12845678) 32, (D12345678) 3244, 4 t, Ar2-g4-t 

[ (d12345678) 321, (012345678)324 4, 4 t, A15-26-37-48-¢ 

[ (a12845678) 325» (D12s45678) 3239-4, 4 #, G16.25-37-48-¢ 

[ (ar2845678)321» (012845678) 3244, 4t, Aias6-t 

[ (G12848678)32,(012845678)32,. 4, 4 t, Asa-s6-t 

[(@ies45678)32,,(012845678)321_-4, 4 t, Ar2-56-¢ 

[ (12845678) 3245 (b12845678) 32914, 4 t, Ar2-56.t 

[ (a12845678)32,, (D12sas678)32,,.14, 4 1735-2806. f, 
Ay2.56-t, As¢. Dy. t, 57.68. 013-04. t, 19.5867 - 013.24. t, 
ig4.57-68- Di4og.t, Ar493-56 - 013.24. 


_IMPRIMITIVE SUBSTITUTION Groups OF DEGREE SIXTEEN 29 


No. 


170 [ (Gresase78)s2,, (Di2sss67s)3215 14, 4 @r7a5-2ss6-t 
171, 172 [(aresss07)s2,, (Drosaso7s)32,, 4, 4, i2-s4-t 
173, 174 — [(auosase7s)s2,,(Di2ass078)32,, 4, 4t, Gi2-s4-t 
175, 182 [(aressse7s)s2,,(Dr2sis678)32,, 4, 4 Q12-56-t, Arse4-s763-t, 
; Gs. Dy. t, Ars-24-5768-D12-t, 1423-56 - Dig-24.t, 
1g4.5768- D1g.24.4, Aig-24-56-Di423-t, As76s- Di423.t 
183, 185 [[(aiasss07s)s2,,(b12345078)30,, 4, 4 t, dre-se-t, 
4625-37-48. 
186 [ (Gross567s)32,,(Dr2345678)3214 14, 4 Q12-34-E 
187, 188  [(ar2s45678)32,,(bi2sss678)321¢_h, 4 Grzseosas- ft, 
12-34-5867 - Digs. t 
189, 191  [(iessse7s)s2,,(Dr2s4s078)32,, 14, 4 t, Gra2s-s867-t, 
Ay 7362845 « t 
192, 195  [((ai2s4s078)32,,(Dr2sise78)30,, 4, 4¢, die-s4-t, 
A423. Di403.t, ige4. Diss. t 
196, 199 [[(arosasezs)s2,,(Dresase7s)52,, 1, 4 t, Ar2-s-t, 
Ay403 - Di4o. t, 1304. Dy403.t 
200, 206 — [(ai2e45678)s2,,(Di2sis678)32,,_h, 4 G12-5768-t, 
Ay5.26:37-48-t, Gyeas-2537-t, 13.24. big.24.t, 
Ay4.23-Dis.24.t, Ar597-2618 - Dis.24.t, 1647-2538 - Dig.24.t 
— 207, 210 — [(aressse7s)s2,,(bi2sse7s)s2,,4, 4 t, die-s76-t, 
(143.24 « 013.04. t, 11648-2537 -t 
211, 214 [[(aiessse7s)s2,,(Di2sise7s)32,,h, 4, Gre-s4-t, 
¢ yg.24. D1g.04. t, 14.93. Di3.24.¢ 
215, 218 [(aiesise7s)s2,,(Di2sase7s)s21,_h, 4 ¢, A13-24-57-68-¢, 
43.24. Dig.o4.t, 57-68 - Dig-24.t 
219, 221 [[(aiasss67s)32,,(Diessse7s)32,, 1, 4 13-24-57-68 +f, 
13.24. D13.24-t, 57-68 - Diz.24.t 
222, 225 — [ (aizs1s07s)s2,, (bi2s1s678)3247_h, 4, 13-24-57-08-b, 
43.24. 013.24. t, 57.68. 013-24. 3 
226, 233  [(auossse7s)32,,(Di2sss078)321,.h, 4 ¢, dio-s-t, 
on y3.24-56-¢, A15-26-37-48-t, Are25-3847-t, Ayzas2ss6-t, 
Pace 1324+ Diseg.t, G1s972648- Dises.t 
234, 236 — [(ar2sis678)2,, (12915678) 32,14, 4 @13-24-56-t, 
304 - Digoq. t 1728-3546 - Digest 


30 


Order 


128 


No. 
237, 238 
239, 240 
241, 244 


245 
246 
247, 248 
249, 250 
251, 252 
253, 254 
255, 256 
257, 260 


261, 264 


4,5 
6, 15 


GrRAcE Marte BAREIS 


[ (a12345678) 3215 (012845678) 3215, 4 t, 

[ (ai2345678)32,, (012845678) 3244 4, 4 t, 

[ (a12345678) 32,5 (12845678) 324,14, 4 t, 
A324. Diz04.t, Gragg. Digoa.t 


y2.34.¢ 
Ayo.34.t 
142.34. t, 


[ (G1o3a5678) 32 is (b12345678) 3246, 4 45.26-37-48-¢ 
[ (a12345678)321, (012845678) 3215_, 4 Ay625.3847 - b 


[ (a12345678)32,, (012845678) 324514, 4 ¢, 
[ (a1o345678)32,,(D12845678)32,, 1, 4 ¢, 
[ (12345678) 3214 (012845678) 3214 14, 4 t 
[ (d12345678) 3214 (012345678)3215_h, 4 t, 
[ (ios45678)32,,(D12345678) 3245 4, 4 t, 
[ (12345678) 32, (Dizs4s078) 3245 4, 4 t, 
Arg24. Digoa.t, A5768- Disos.t 
[ (a12345678) 32, (012345678) 32151, 4 €, 
A304. Digoa.t, As768- Diso4.t 
[ (12345678) 9619012345678) 96,9_h, 1 4, 
[ (12345678) 96 (012345678) 961, 1 
[ (ar2sas678) 96, (Di2348678) 96,,-h, 1 4, 
[ (a12345678) 96,, (D12345678) 96, h, 1 f, 


13-24-56 + t 
13-24-56 + t 
(113.24-56 + t 
14.23.56 +t 
(11324-5867 - t 
(304.5867 +t, 


(1324.5867 « t, 
42-56 + Dy2.56 a 


12-56 - bio.56 .t 
(2.34-56-78 +E, 


Ay. Dip. t, Aga.56-78-D12-t, G1g.24- big.24.t, 


114.23-56+78 « bis.04 . t, 1493 - by428 . t, 1324-56-78 + by493 .t ; 


[ (a1o345678)48;,(D12345678)48,,_b, 2 ¢ 
[ (aizs45678)48,, (b12345678)48)_b, ot 


[ (12345678) 24, (Diosase78)245 4 , 4 015.06-37-48-t, 


116.25-37-48 - 019-56. t 
[ (auosi5678) 8, (D12s5678) 855 | # 
[ (12845678) 894 (012845678) 89¢_| ¢ 
[ (a12845678) 8) (012345678) 84» | t 
[ (auosis67s) 8,,(D12345678) 8», 1 £ 
[ (ai2345678) 8,5 (012845678) 895_] t 


[ (d1345678) 169, (012845678) 1697_l4, 4 t, C1256, 


Qise4.5768-t, A13-24-57-68-b 
[(Gizs45678)1697 (B12845678) 165,14, 4 t 
[ (ayo345678) 169, (D12845678) 1697 lt, 4 ¢ 


IMPRIMITIVE SUBSTITUTION GROUPS OF DEGREE SIXTEEN 


Order 


No. 


12 
13, 16 


17 

18> 

19 
20, 23 


[ (@12845678)16,,(D12345678)16,,_h, 4 t 

[ (@12345678)16,, (D12345678) 16,1, +t, A12-56-t, 
y5.26-37-48-¢, A16.25-37-48-t 

[ (a2345678)16,, (Dr2s4s678) 16,,_14, at 

[ (ai2345678)16,, (D12845078)1657 1, 4 t 

[ (ai2845678)16,, (012845678) 165, 1, 4 € 

[ (aizs4s678)16,,(Di2aise7s)16,, 14, 4 t, Ar3-24-87-68-E, 
15.26-37-48-t, 15.26 - b15.26. 

[ (Gi2s45678)16,,(Dr2845678)16,, 1, 4 Qrse4-5768-¢ 

[ (42345678) 165, (D12345678)1697 14, 4, A15-26- 015.26. €, 
2.56+ t, 17.28-35-46-¢ 

[ (Gizss5678)16,,(D12845678)1697 14, at 

[ (@ies45678)16,, (D12345678)16,, 4, 4 t, Aie-s6-t, 
115.26-37-48-¢, A16.25-37-48° € 


[ (Giesas678)16,,(b12345678) 165,14, 4 t 

[ (ai2345678)16,,(D12845678)16,, 4, 4 ¢ 

(12845678) 16», (b12345678) 16)7_14, at 

[ (Gizs15678)16,,(Pizsis078)16, 4, 4 t, ro-s6-t, 
1625 - Digos - t, 143-24-57-68-¢ 

[ (Gi2s4s678) 16), (P12345678) 16, 14, 4 f 

[ (@iosss678)165,(12345678)16,, lk, 4 t, Are-56-t, 
43-24-57-68-1, 1423-5867 -¢ 

[ (ar2g4s678)16,, (012345678) 16), 14, 4 ¢ 

[ (@2345678)16,, (12345678) 16,, 14, 4 ¢ 

[(@r2s45678)16,, (Di2s4s678)16,, 14, 4 ¢ 

[ (a12345678)16,, (b12345678)16,, 4, 4 t, 13-24-57-68-, 
17.28-36-45+t, 1625+ D165. 

[ (Gie345678)16,, (D12845678) 16,, 1, 4 ¢ 

[ (Gioss5678)16,,(Di2s4s678)16,, 4, 4 t, Are-s6-t, 
y625-3847-t, y526-3847-t 

[ (Gi245678)16,, (b12345678)16,, 14, 4 ¢ 

[ (@i2345678)16,,(Di2s46678)16,, 4, 4 t 

[ (ar2s45078) 16,, (D12345678)16,, 4, 4 t 

[ (Gios45678) 165, (Diesis67s)16,,.4, 4 t, Aie-s6-t, 


625-3847, 1526-3847 -¢ 


31 


32 


Order 


GracE Marie BaRreEIs 


No. 


65 L (a12345678)16,, (D12845678)16,, 4, 4 
66 (412345678) 16), (D12345678)165, lt, 4 t 
67 [_(a12345678)165, (012345678) 16), 4, 4 t 
68, 69 —— [(Gi2345678) 165 (012845678) 1695 lk, 4 @1735-2846. €, 
157.68 - Dy3.04.t 
70, 73 [ (12845678) 16,5 (012845678)1605 14, 4t, 2.34.1, 
12.56 - Dio.56.t, Ai735-2816.t 
74 [ (@12345678) 165, (D12845678) 169, 14, 4 ¢ 
75, 78 | (diz345678)16,4(b12345678) 169, 4, 4 t, Aro.a1-t, 
113.24. Di3.04.t, A14.28- Dig.04.t 
79, 82 (12845678) 16, (012845678)1654, 4, at, Ayp.34.t, 
3304. Dig04.t, Ar7sso845.t 
83 [ (a12345678) 1655 (D12345678)16, 1, 4 t 
84, 85 | (d12345678) 1654 (12345678) 1695 14, 4 A124 Digos. t, 
Cr7zez8i5.b 
86, 89 — [(ar2845678)1655(P12845678)16,, lk, 4 t, Gi2-56- dio-s6.t, 
134.56-Di2.56-t, Gr7362845-t 
90, 93 — [(ar2845678)164(B12345678)1659 4, 4 t, @13-04-56-E, 
(115.26-37-48-4, Aiza5asa6-t 
94 [ (d12845678) 16, (12845678) 16,9 4, at 
95. (412845678) 1655 (12845678)16,_l4, 4 t 
96 [ (G12845678)16,)(012345678) 16554, 4 f 
97, 100 — [_(@12845678)1655(B12845678) 16514, 4 t, Aig.04.56-¢, 
(1625-3847 -¢, Cisase736.¢ 
101 [ (12845678) 1695 (012845678) 1699 le, 4 t 
102 [ (12845678) 16) (12345678) 16) l4, 4 t 
103 [ (12345678) 169 (012845678) 169 14, 4 f ; 
104, 105 — [_ (ai2s45678)3249(Dios4s678)3210_h, 2 t, Gio-s6 Di6.25-37-48-¢ 
106, 109 — [_(ai2s4s678)30,,(b12945678)324, b, 2 t, A13-24-57-08-E, 
y2.34.t, A14.93.57-68-¢ 
110 [ (ai2s45678)32,, (Di2sas678)32,,_b, 2 t 
111 [ (a12345678)32,, (D12345678)32,,_b, 2 t 
112, 118. (12845678) 321 (B12345678)3239_b, ot, 13.24.57-68-t, 
Ay304.5768-t, Gig.34-t, Aig. Dig.t, Gig.c4. di3.24.t, 
1423 « Dy493 oe 4 


» 
= 
be 
." 
Nee 
a) 
i 
— 
‘* 


Order No. 
— 119, 120 
121 
122 
123, 124 
125 
126, 128 


129 
130, 135 
136, 138 

_» 139, 142 

Bee fg 148, 146 


147, 150 


159 

160 
161 
See 
163, 1 


Se £168, 176 


151, 154 


155, 158 


465, 167° 


ImprimitivE SupstiruTion Groups oF DrGREE SIXTEEN 


[ (aie345678) 3215 (012845678)3245_b, 2 ty Gis-24- Dis.os.t 

[ (ar2s45678) 321» (b12945078)32 1b, 2 t 

[ (a12345678) 2,5 (Dr2s45678)321,_b, 2 t 

[ (arzss5678) 324» (Orosase7s)324..b, 2 t, r4es- Disos.t 

[ (arogss678) 321» (O12345678)321_k, 2 # 

[(a2345678)32;4(D12345078)3219 b, 2 t, Aiss-246-biss.264-t, 
153-264. Org5-246-6 

[ (azp4s678) 521, (Di2s4s678)321_b, 2 t 

[ (Giess5678) 32,5 (012845678)324-b, 2 t, Aia-s4-F, 
Ayo.56-t, 143.04. Dy3.94.t, 157.68 - O13.04. 1, 
1597-2648 « O13-04.t 

[ (ai2s45678)32)»(Di2s4s078)324.b, 2 t, Aie-sa-t, 
y3.24-57-68-¢ 

[ (ai2815678)32 1, (D12845678)32,. b, 2 t, Ai2.s4-t, 
y3-24.57-68-t, 14.03-57-68-¢ 

[ (ai2sase78)32,,(Di2845678)3219-b, 2 t, Ai2.56-t, 
Gye25-3847-t, Ar526-3847-¢ 

[ (aizs4s67s)32,»(Di2s45678)32,._b, 2 t, Ar2.56-t, 
5.26-37-48+t, G16-25-37-48-¢ » 

[(a12345678) 32, (O12818678)521_b, 2 t, A13-24-57-68.f, 
y5.26-37-48-t, G17-28-35-46-E 

[(a12845678)32,.(b12345678)321. bk, 2 t, A13.24-57-68-t, 
(yo25-3847-t, is27-3045-¢ 


[(Gies4s678) 32,» (Di2sis678)32,,_b, 2 t 


L(esss678)325 (biesss67s)s2,,_b, ot 


[ (a12815678)32 1, (D12345678)32,, b, 2 ¢ 
[ (ar2345678)s2,4(012915678)325 b, 2 t ” 
[ (ai2ss6678)325(b12345678)3215-b, 2 t, 33-24- Dis.o4-t 


‘L(@2s4s078) 3219 (bi29ss678)3219_b, 2 t, Ar35.246 - Diss-264-t, 


A304. Dy40g.t 

[ (aiosss678)32,,(Oi2s4s078)324y-h, 2 t, Are-sa-t, 
15.26-37-48-t, Gheos.ss17-t, Mie. Dio.t, dag. dy.t, 
045.26 -Dj5.06-t, A625. Dre2s.t 

L(@ies8s678)32,,, (b12315678)321, b, 2 t 


33 


34 


Order 


112 


No. 
177, 179 


180 
181, 185 


186, 190 
191, 193 


194 
195, 196 


197, 200 
201, 202 
203, 204 
205, 206 
207, 210 


211, 212 
213, 216 


Grace Marre BAREIS 


[ (12845678) 3215 (Di2s45678)321,.2, 2 t, Mie-s4.t, 
115.26-37-48 +6 

[ (a12845678) 3245 (D12845678)3214-b, 2 t 

[ (duzsas678) 32 ie (Diosts678) 3215.2, 2 t, Aie.s4-t, 
117.28-35-46-t, Gige7-3645-t, A143 Dy4o3.t 

[ (dios45678)32,,(Dr2845678)321, b, 2 t, Aio-s4-b, 
115.26-37-48+t, 13.24-D13.04.t, A1625-3847-t 

[ (12845678) 321. (012345678)3212 2, 2 t, Ai2.34-t, 
116.25-37-48-¢ 

[ (12345678) 3215 (D12345678)3215 2, 2 & 

[ (a12345678)32,, (D12845678)32,5_k, 2 413-24-57-68-E, 
57-68 + Dig.o4.t 

L (12345678) 325 (Di2sss678)3215_b, 2t, dye.34.8, 
(113.24-57-68-£, G14.23.57-68-¢ 

[ (ai2345678)32,, (Diosas678)32,, 2, 2 G12-56-t, Ose. Dio.t 

[ (di2s45678)32,, (0123456783214 b, 2 Q12.34-t, Agai56 

[ (12345678) 32,;(bi2845678)32,, 12, 2 A1423-s867-t, 
12.34.5867 + Digg. t 

[ (di2s45678)32,, (D1234567892,, b, 2 t, Gie.sa-t, 
(1423-5967-¢, M1324-5867-¢ 

[ (ar2sa5678)32,,(Dies4s678) 324, 2, 2 t, Ais.24-Dis.04-t 

[ (aiz345678) 32,4 (012845678) 3214-2, at, Ay2.34.t, 
(11324-5867-t, 1423-5867 -¢ 

[ (a12345678) 321, (D12845678) 32452, 2 t 

[ (@12345678)3245 (012845678) 3215_k, at 

[ (di2345678)56(D12345678) 56h, 1 

[ (ar2sa5678)48,, (012345678) 48,4_h, 1t, y5-26.37-48-8 

[ (12345678) 48,4 (012345678) 48;_h, 1t, G15.26-37-48-¢ 

[ (ar2345678)48,, (Di2s45678)48,7_h, 1t, Qio.34-56-78-0 

[ (ioa5678)48,, (D12345678)48,,_h, 1 t, A12-34-56-78-t 

[ (a1o345678)24, (D12345678) 249 b, 2 t, Ar2-56- Di2.56.t 

[ (a12845678)24;5(D12345678)2449_b, 2 t, di3-24-78 - b1s.24-73-t 

[ (Gr2s15678) 895 (12845678) 89,14, 4 t, is-26-37-48-¢ 

[ (Grogi5678) 895 (O12345678) 8,5 4, 4 Cise4-5768-t 


IMPRIMITIVE SUBSTITUTION Groups OF DEGREE SIXTEEN 35 


Order 


No. 


[(@i2345678) 85, (012345678) 89h, 4 t, Grs26-3847-¢ 

[(arzgss67s) 8,o(012345678) S95 ht, 4, Cizsersas.t 

[ (aresis678)8,,(D12915678) 8.4 14, 4 @15-26-37-48- 

[ (aross67s) 8,,(Dr2345678) 8,4, 4 ¢, Aises-s768-¢ 

[(ar2g45678) 8, (12345678) 855 ht, 4, 15-26.7-48-t 

[ (a12845678)165 (012345678) 165,22, 2 t, Arses-5768-t, 
Qy2.56-t, A1g.24-57-68-4, 15-26-37-48-t, M116-25-37-48-E, 
45.26. 045.26. t, 46.25. 015.26. t 

[ (a12945678) 165, (012545678)165,_k, 2 t, Grse4-s768-, 
Ay.56.t, 43.04-57-68-¢ 

[ (Gr2s45678) 16,,(D12345678)16,,_b, 2 t, Aises-s7es-t, 
Ayo.56.t, 13.24-57-68 +t 

[ (ai2345678) 16, (012345678)16q,_b, 2 @is24-5768-t, 
Ay2.56.t, 143.04-57-68-¢ 

[(@i2845678)165,(D12s45678)16,, b, 2 t, A15-26-37-48-¢ 

[ (a12345678)169, (012945678) 169,_k, 2 t, A16.25-37-48-¢ 

[ (aizss5678) 16), (b12345678)165,_b, 2 Qise4-5768-t 

[ (a12845678)16,,(012845678)16,, _b, 2 t, Grse4-5768-¢ 

[ (au2gs5678) 165, (Or2345678)16,,_b, 2 t, Arges-5768-t 

[ (ar2345678) 165, (biesis678)16,;_b, 2 M15.26-37-48-¢ 

[ (ai2845678)16,,(012945678)16),_k, 2 t, @15-26-37-48-¢, 
45.26. 015-26. t, 016.25 015.26. t 

[(Gies45678) 16,, (b12345678)16,,_k, 2 t, 12-34-56-78-¢ 

[(ar2s15678)16,,(012945678)16,, k, 2 t, Gi2-s6, 
Q13.24-57-68-¢, Gr423-5967-t, Gre25-3847-t, 
Ars26.3847-t, Aye25- Die25.t, Ars26- Die2s.t 

[(ai2915678)16,,(D12845678)16,,-k, 2 t, 13-24-57-68-¢ 

[ (ai2945678)165, (012845678) 16,, 2, 2 t, di2-s6. 

[ (ai2s4s678)16,, (D12845678)16,,_b, 2 t, dio-s6-t 

[ (ai2s45678)16,, (D12545678)16,, b, 2 “1625-3847. t, 
32.56.3847 + Die25.t, Are25- Drees. t 

[ (aross67s)16,,(O12345678)16,, b, 2 t, re25-sea7-t 

[(Giesss67s)16,, (D12345678)16,,.b, 2 t, ise6-s47-t 

[ (a12845678)16,,(D12846078)16,,.b, 2 t, Cro2s-ss47-t 


36 


Order 


48 


No. 
Tigt2 
73, 76 


77, 78 
79, 81 


82, 83 
84 
85, 86 
87, 88 


89, 90 
91, 92 
93, 94 
95, 96 
97, 98. 
99, 102 


103, 106 


107, 108 

1, 4 
“3a G 
7, 10 


Grace Marte BargEIs 


[ (a12845678) 16, 5 (b12345678) 16, sb ,2¢, Aise6.3847-t 

 (a12345678) 16,, (D12345678) 16, 2, 2 t, 13-24-5768. ¢, 
1596 - Digas. t, Aigese457 « Die25. t 

[ (12345678) 16,; (D12845678)16,, b, 2t, (13.04.57-68-¢ 

[ (a12345678)16»5 (012845678) 1654 2, 2 1735-20846. t, 
157.68 - b13.04.t, 58.67 b13.04.¢ 

[ (ai2s45678)16,,(D12845678) 165, 2, 2 t, Gr735-2806-t 

[ (a12345678) 1655 (D12345678)169, b, 2 @12-34-56-78-E 

[ (a12345678)16, (012945678) 1695 bk, 2 t, Arz.34-56-78-¢ 

[ (d12845678):169, (b12345678) 1694) , 2 A17362845-t, 
12.57.68 » 013.2456 t 

[ (a12345678)16), (012845678) 16, P, 2 t, Arzaezsas-t 

[ (12345678) 16, (D12845678) 16552, 2 t, Ars24.5867-¢ 

(12845678) 16, (012345678) 16,5_k, 2t, Ay304.5867-¢ 

 (a12345678)1695 (012845678) 169), 2 t, Grse4-5867- 

 (di2s4s678) 16,4 (B12845678) 16,9 b, 2 t, Arse4.5867-E 

[ (a12s45678)32,,(B12345678)32,, h, 1, Qt2-34-56-78-t, 
@13.24.57-68-t, Gi4.23-58-67-¢ 

[ (a12345678) 325 (D12s45678) 32,1, 1 t, diz-34-56-78-f, 
043.24 Di3.24.t, A14.03-56-78 - 013.04. ¢ 

[ (ar2345678) 321, (012345678) 32,, h, 1 t, Qr2-34-56-78-¢ 

[ (auesss678)24, (Dros1s678)24, hi, 1t, 15.26.37-48-€, 
0116.25-37-48- O19.56-t, Gr2.56- Dio.56.t 

[ (ar2s45678)24,,(Di2s45678)24,, h, 1 t, A12-56- Di2.56-t 

L (ai2845678)24;5(D12345678)24)9_h, 1 £, Ai2-34-56-78- ¢, 
13.24.78 + b13.04.78.t, 14.03.56 - D13.24-78-t 

[ (a12845678) 8), (D12845678) 8, _b, 2 Aise4-s768-t 

[ (12345678) 8,» (012845678) 8), 2 @154s2637- t 

L (412845678) 165, (D12845678)16y,_h, 1, Gi2-34-56-78-f, 
115.26.37-48-t,. 19.34. Di2.34.t, A15.26- bi5.26.t, 
37-48 D15.06.t, A1526-34- Dis06.34-t, 1625-78 - D1506.34. 6 

[ (a12345678)1697(D12345678)16,,_h, 1, Qie-34-56-78-t, 
15.26-37-48-t, A16.25-38-47-¢, 15.26. 015.26. t, 
A16.25-34-78- 015.26.t, 37.48. 015.26. t, 
1256-38-47 « Di5.26.t 


IMPRIMITIVE SupsTiITuTION Groups OF DEGREE SIXTEEN 37 


Order No. 
<— 19, 22 — [[(aros4567s)16,,(b12845678)16j,_h, 1 f, 12-34-5678. ¢, 
45.26-37-48-t, A19-34- Dr2.34.t 
23, 29 —_ [(arasaso7s)16,,(D12345678)16,,_), 1 “1625-3817-t, 
Gs6.78- Di2.s4-t, Are25-Dieo5-t, 1526-34-78 - Oreos. t, 
Ag74g- Dig25-t, A15-26-34- D15.26-34-t, 
46-25-78 - b15.26-34.t 
30, 34 [ (aie34s678) 16, (012845678)169,_), 14, Gy2-34-56-78-t, 
yp05-3847-t, Gr2.34-Dr2.34.t, Gs6-78- Dio.s4.t 
35, 40 —_ [(a12845078)1695 (D12345678) 1655), 1 12-34-56-78-t, 
13.24. Dis.o4.t, G14-23-56-78- Dig.04.t, 57-63. diz.04.t, 
12.56 + Dio.56- t, 14324-56- Dy 304-56 oF 
41,48 — [(aisss678)16),(012845678)1695_h, 1 f, A12-34-56-78-¢, 
G43.04.57-68 +t, A14.23-58-67-¢, A1g.24- Diz.04-t, 
14.03-56-78 + Dig.24-t, 57-68 - O13-24-t, 
42.-34+58-67 + Di3.04 ey 
49,55 — [(divsase78)16,(b12345678)16,, 1, 1 ises-s7e-t, 
Ag4.73- Drg.56-t, 14-23-78 - D13-24-56-t, Aige4. Disos.t, 
(112.-34-5768 + Drz04 ° t, 142-56 - Dyo-56- t, 
, 3-24-56 - 013-2456. ¢ 
56, 59 — [(ar2sss678)1645(Dr2345078)16,_h, 1 t, A12-34-56-78-t, 
Rs Gy2.34- Dio.g4-t, A56-78- Or2.34.t 
16 as [ (Gres4s678) 8,5 (D12345078) 8), _h, 1 ¢, Ar2-s4- Dro.g4.t, 
9.56. 019.56. t, 115.26 - O15.06 - t, 34.78 - Di2.56. t, 
15.26-37-48-t, 16-25-37-48 - Or2.56.t ; 
8,13 — [(aisss67s) 895 (D12345678) 89, h, 1 f, Ar2-34- Dio.s4-t, 
G13.24-56- D1g.24-56-t, G1sase637-t, 14.23-78- big.24-56-t, 
44-23-56» Dy4.23-56 .t 


14 [(aies45678) 8,, (D12345678) 8), _h, 1 ¢ 


* 


SYSTEMS OF INTRANSITIVITY OF Heaps 4, 4, 4, 4. 


Order No. 


7962624 ] [ (a1234) all (by34) all (C1234) all (dyo34) all ] t t ts 
3981312 ] [ (ara) all (by234) all (C1934) all (di234) all ] t ty 
2,3 L{ (Giz34) all (Diz34) all (Cro34) all} (dios) all Jesi2, 12 
tt ts, t ty Ay. tz : 
1990656 1 [{ (1234) all (by234) all (C1234) all} (di234) 
all lore, 2 th 
995328 1 ; 2 [ (ai234) all (by34) all (C1934) all (dio34) 
all jo, 2,2, wth ts, de.ce.th dy.t 
497664 2 [ (aioz4) all (by934) all (e134) all (diag) 
all ho, 12,12, 2th, dy.¢y.th 
331776 1 [{ (A234) all (by234) all (C1234) all} io, 12, 12 (dio34) 
all |sze, 4 t th ts 
248832 1 [(ai2s4) pos (biz34) POs (C1234) POS (diss) pos | t th 
165888 1 , L{ (Gizz4) all (Diz34) all (C134) all }io, 12, 12 (dias) 
all |sz6, 4 tt 
2,4 [{ (G1234) POS (1234) POS (C1234) pos} (dizsa) 
pos |s76, 4 t t ts, tty Ay . bye Cio. dy ‘ ts, 
t dyo3. ty ia. by. C12. dig. ts 
98304 1 [ (1234) 3 (Dizss) 8 (Crzs4) 8 (Aizsa) 3_| tt ts 
82944 1,2 [(aies4) pos (br234) POs (C1234) pos (dr234) 
pos |sve, att, t dvs.t 
49152 1 [ (diese) 8 (bioss) 8 (C1234) 8 (diosa) s | t ty 
2,3 [ { (aizss)8(Diz24) 8(Ci2s4)8} (dioss)s_bss, 4 t ty ts, 
t ty a2. ts 
4,5 L{ (dress) 8(Di234) 8(Cios4) 8} (diosa) s_bs6, 4 t th ts, 
t i 113-24. ts 
6, 7 L { (@12s4) 8 (D134) 8 (C134) 8} (dioss)s_bse, 4 t tr ts, 
tty 2. ts 
; 36864 1 [{ (Ay234) all (by234) all (C1234) all}, 4,4 (d1234) 
all Jes, 4 t ty ts 
38 


IMPRIMITIVE SUBSTITUTION GROUPS OF DEGREE SIXTEEN 


18432 1 


12288 


9216 


6144 1 


[{ (Gizss)s(bies4)s(C1234)8} (i2ss)s_bso, 4 t th 

[{ (aos) 8 (1234) 8 (C1234) 8} (diss) s_bso, 4 t th 

[{ (a1os4)s(Dr284) 8 (C1284) 8} (diese) s_bse, 4 t hh 

[ { (124) 8(bies4) 8 (Cress) 8 } (diosa) s hes, 2 t tr ts, 
th dig. tz, tt, dy3.04. tg, th dy304. ts 

[{ (areas) all (Dress) all (Cross) all}, 4, 4 (diosa) 
all les, att 

L { (aiesa) pos (Dioss) POS (C1234) POS}4, 4, 4 (divas) 
pos ls, attr ts, tt die. di. ee. dig. ts 

[{ (dross) (Dress) s(Ciess)8} (dross) s_hes, 2 # tt 

[{ (aio3s)s (Dios) 8(Ciess)s}7, 4, 4(dios) les, 4 t th ts 

[ { (diese) s(Drioss) s(Cioss)s fa, 4, a(dioss)s les, a t th te 

[{ (a4) s(bios4)s (Cross) s}4, 4, 4(dioss)s los, 4 t th ts 

[{ (ayo) all (Dyo34) all (1934) all}, 4,4 (dyo34) 
all he, 1th ts 

L{ (ay234) POS (D124) POS (C1234) POS}4, 4, 4 (dio34) 
pos la,ath — 

C{ (dizss)s (Diss) s(Cross)3}4, 4, 4(dioss)s los, 4 t th 

[{ (Gress) 8(bioss) 8(Cioas)s}4, 4, 4(dioss)s los, 4 tt 

[{ (Gu2ss) s(bioss)s(Ci2ss)8}4, 4, 4(dios4)s los, 4 t th 

[{ (ai2ss)8 (D124) 8(Ciosa)s}4, 4, 4(dr2ss)s_be, 2 t th ts, 
tt i304. ts, Ay. diy.t ty dy. tz, 
ay. diy .t ty 13.24. ts 


[{ (aioss) s(Dizss) 8 (C1234) 8}4, 4, 4(Aioss)s_bse, 2 t th ts, 


tty dy. ts, Ar3-24- di3.04. ty dis.24. ts, 
113.24 - d13.04.t ty digo4. ts 
[{ (Gizss) s(Dioss)s(Cias)s}4, 4, 4(diess)s be, 2 tt fe 
tt diz.o4.ts, die. dig.t tr dio.ts, 
yp «dy. t ty disos. ts 
[ { (@izgs)s(Dioss) 8 (Cros4)8 $16, 2(Aizs4)8_ he, 2 t th ts 
[(ai23s) eye (brass) eye (C1234) eye (diss) eye ] 
t i ts, t t; Ajo. Dyo. yo. dig. ty 
[ (ar234)4(Dr2s4)4 (C1234) 4 (dross) | t tr ts, 
t ty yy. bya. C19. dhe. te, t Ayog . D439. diog. ty 
Aya. bio. Cog. dos. ts 


39 


40 


Order 
4608 


3072 


2304 


1536 


Grace Marre BAREIS 


L{ (Grogs) all (bio34) all cis) all}s, 4, 4 (diosa) 
all he, 1 t t 

L{ (dias) pos (biess) POs (C1234) POs}4, 4, 4 (dross) 
pos lis, ut t ts, t i ay. Dye Cio. diy ‘ ts 

[{ (Gizsa) 8 (Disa) s(Crosa)s 4, 4, 4(dizz4)s he, 2 tts, 
ay. dis re ty 

L{ (124) 8(Biose)s(Cioss)8}4, 4, 4(diosa)s be, 2 tt, 
(43.24 - dhs.o4.t ty 

[ { (aizsa) (D234) 8(Cisa)8}4, 4, 4(dio34)s be, 2 th, 
ay. dig st ty 

[ { (Gios4)s(Diose) 8(Cros4) 8 }16, 2(dizsa)s_ be, 2 th 

L{ (diss) s (bres) 8 (Ci2s4) 8 }4, 4, 4(diesa)s_hs, 1 tt ts, 
t ty dyo.g4. ts 

[{ (Grose) 8 (Disa) 8(Cizsa)8}4, 4, 4(dizs4)s_he, 1 tA ts 
t ty dio.34. ts 

L{ (dio34) cye (bi234). eye (Coss) Cyc} (diosa) 
eye |tth 

[{ (aios4)4(Dio34)a (C1934) 4} (dioga)a | is 
t digg. digo. dios. t 

[{ (disse) eye (bizs4) eye (Cress) eye} (diosa) 
eye he, 2th tz, th dj304. tz, tt Mg. dy. Cyn. dy. tg, 
t ty a2. bre - Cig. dy3.04 . ts 

[{ (Gizss)4(Diose)4(Ciosa)4} (diess)4 fe, 2 t th ts, 
t t Ao. Dig. Crp. dig. tg 

L{ (di234) POS (bi234) POS (Cros) POS}a4, 4, 4 (dyo34) 
pos he, 1 t ty 

C{ (ias4) 8(Dioss) 8(Cizsa) 8 }4, 4, 4(dioss)s he, 1 tt 

[ { (Goss) 8(bios4) 8(C1oaa)8}4, 4, 4(diosa)s he, 1 t th 

[{ (ar2s4)8(bioss) 8(Cizs4)8}4, 4, 4(diosa)s_he, 1 tt 

L{ (d1234) 8(D1234)8(C1234)8}2, 2, 2(dizaa) sls, 2 
Aig. Cio. t ty dia. ts, 13.24. C13-24-t th dig.o4. ts, 
Aigo4. Cigoa-t ty dizoa. ty 

L{ (aros4) eye (Dies) eye (C1234) eye} (dizsa) 
cyc he, oth 


IMPRIMITIVE SUBSTITUTION GROUPS OF DEGREE SIXTEEN 41 


Order No. 
8, 11 


13, 17 


1152 1 


768. 1:3 


11, 13 


576 1 


C{(ai2ss) cye (brass) eye (Cross) eye} (diss) cycles, 1 
t th ts, t th dyy.34. ts, t t Ay. by. Cro. dyy. ts, 
th Ayo. Dyo. Cre. Ayg.04. ty 

[ { (ai2ss)4(Di2ss)4(Cr234)4} (di2s4)4 be, 2 t th 

[ { (Gizss)4(Di2s4)4(Crosa)a} (dieas)s he, it tr ts, 
t t dyo.34. ts, t ty Ay . bio. Cro. dip. ts, 
t yg - Dy93. Crog. digg. ty My. bio. Cro. dig. tg, 
dyy.34.t Ay93 . Dy93.. C193. digg. ty Ayo. Di. Cp. dig. ty 

C{ {(aias4) pos (bies4) pos}, 4 (Cross) POs}, 1 (dross) 
pos |, 1 t ty Ayp. Dye. Crp. dig. te 

L{ (i234) s(Diess)s(Cross)s}2, 2, 2(dioss)s Js, 2 
Ay. Cig.t h, 143.24. C13.04.¢ hi, 3304. Cizoa. t ty 

Cf (3234) 3(Di234) 8 (C1284) 8 }2, 2, 2(di3s)s_|s, 1 
3.04. Dy3.94.¢ ty ty, Aig-24. Dis.04.8 th dio.ga. bs, 
Ay. byt t ts, Ay. by. t t dyp.34. ts, 
1304 . 01304. dyo.34.t i ts, 1304. Dy304- Aya.g4.8 t 
diz.34.ts 

C{ (dias) eye (Diese) eye (crass) eye] (diosa) 
eye he, ith 

L{ (aiza) eye (biess) eye (crza1) eye}s, 2, 2(dies) 
eye |s,2t th diz. bi.¢r. die. ts, 
Qy304 - Ci304.¢ t dizo4. tz, 4304 - C1z04.t t 
Aya . Dy. Cro. . di3.24. ts 

[ { (dioss)s(ros4)4 (Cross) 4} (diosa) he, 1 tt, 
t dyo3. i193. C193. digg. th, dip.34.t 
Qy93 . Di93.. C193. digg. 

L { (@rzg4)4(Dioss)4(Croa4)4}2, 2, 2(dioss)a Js, 2 
th Ay. bia. Cre. dy. tz, 43.04. Cig.24.¢ th dhs.24. ts, 
y3.24- Cig.24.8 ty Aya. bro. C12. digas ts 

C(arza4) all (Dioss) all (cross) all (dizss) 
all, 1,1, 1th ts 

L{{(aisas) pos (bis) pos}, 4 (Cisss) POS}, 1 (dizs4) 
pos kith 


288 


192 


144 


96 


11, 12 


13, 17 


2,3 


3, 6 


GrRAcE MARIE BaREIs 


[ { (1284) 8 (bi284) 8 (C1234) 8 bo, 2, 2(diaga) 3 lt oe 
- Ai3.24. D13.24.8 th, A13.24. diz.oa.t th, 

A304 -dy493.t ty, Argea. Dios. dig.sa.t th, 
bisea. digog.t th, dig.o4. diz.o4.t t 

L{ (@y234) cye (biog) Cye (Cress) CyC}o, 2, 2 (diosa) 
Cyc |g, 2 Gi304. Cige4. t th 

[{ (di2s4) eye (brass) eye (Cress) cye}o, 2, 2 (diese) 
Cyc |s, 1 Gizea. diayg.t th dig. dig. C1. di4.o3- ts, 
C1324 + dy4og.t ty dip. Di. Cre. diz.o4. ts 

L{ (A234) 4(Dioga)4(Crosa)4}o, 2, 2(dies4)4 ls, 2 
(13.24. Cig.o4.€ ty | 

L{ (Aio34)4 (Diosa)4(Ciosa)a he ,2,2 (diosa) a, 1 
13.04. dy3.04.t ty dig.o4. ts, 13.24. dig.24.t t dis.o3. ts 

L { (Gizss)4(Diosa)4(Ciosa)4 4, 1(diosa)a_, 1 t th 
Aya. by. Cra. dig. tz, t Aig. dig. C123. digg. th 
ye. bia. 1g. dig. ts, t dies. biog. Crag. digs. th 
1493 - Dye - CQ. dip. ts, t 132 - Dizo « C139 « diz9 Gh 
ayj9. Dro Ci. dy : ts, t 1439. Dizo - C39. diag . t 
a3. Dis - C13. dig ° ts 

L{ (Grog) all (Diz3s) all (Cizg4) all}i, 1, 1 (diosa) 
all ji, 1 t ty 

L{ (A234) POS (Di234) POS (C1234) POS}1, 1, 1 (diza4) 
pos h, 1 t t ts, t th M2. Dio. Cy. dip. ts 

L{ (digg) Cye (Dress) cyc (C1o34) cyc}o, 2,2 (diz34) 
cyc |s, 1 Gizeq. dig. t th 

L{ (934) 4(Dios4)4 (Cros) } 2,2,2 (diosa) ls ail 
13.04. dy3.04.t th 

L{ (A234) 4(Dio34)4 (C1234) } 4, 1(dioga)4 lt, 1 
t i, t 1193 - byo3 . C193. dio3 _ h, t 432. Dizo . C139. diz9 > h, 
t 32. brs. Crise. diag. th 

L{ (dio34) pos (Dios) pos (C1234) pos}i, 1,14 (dios) 
pos hi, "y t t - 

L{ (dies) eye (Divas) eye (Czas) Cye}1, 1, 1 (diese) 
cyc h, 1 t ty a2. Dr Cio. diy : ts 


Supstirution Groups or Ducrer Sixrren 43 


2,6 — [{(arass)s(bioss)s(Cross)a}1, 1, 1(diesa)a hr, 1 
th ay. b.c.dig. ts, t dies. bies. C123 - dias «th 
aa. big. Cra. diets, dss. bizs . C123 dies. th 

——— Arz24. Disag. C12 daa ts, Aia.2a- bis.24. die-a4.t 

Ay. bias. C123 - deg. th dhe. bie. C12. diz. bs, 

144.03 « D13.24- di.34.t G23. bi23. Ci23 - dias. th 

i304 - brgos . Cro. daa. ts 

L{ (azss)4 (Diese) (Cross)4}1, 1, 1(dross)a h, 2 

t dog. Dyes. C12. dios.ty, Ar4.2 « bis.24. di2.s4-t 

dyes - Diag. C193. digs. ty 


SYSTEMS OF INTRANSITIVITY OF HEaps 2, 2, 2, 2, 2, 2, 2, 2. 


Order No. 


10321920 1 [ (ai) (biz) (C12) (diz) (€12) (fiz) (giz) (a2) ] ti tr 
5160960 1 [ (az) (biz) (C12) (dha) (€12) (fiz) (gre) (haz) ] th te 
2,3 [(aiz) (biz) (¢12) (diz) (€12) (fiz) (912) (Aaz)_| pos th tr, 
t dye. ty 
2580480 1 [ (az) (biz) (12) (diz) (€12) (faz) (912) (Aiz)_] pos th te, 
344064 1 [(a12) (biz) (C12) (diz) (€12) (fiz) (gta) (a) | th te te? ts 
172032 1,2 [ (a2) (biz) (C12) (dhe) (€12) (fiz) (gta) (hte) ] 
pos ty ty ts? ts, ty ts ts” M42. ts 5 
86016 1 [ (aie) (biz) (C12) (diz) (€12) (fiz) (giz) (iz) ] th t ts 
43008 1 [ (az) (biz) (C12) (diz) (€12) (fiz) (gra) (Az) _] th te? t 
2,3 [(aiz) (biz) (C12) (diz) (€12) (faz) (giz) (faz) _] pos ti t bs, 
ty dg. te die.ts 
21504 1,2 [ (a2) (biz) (C12) (diz) (€12) (fiz) (giz) (haz) _| pos th te #2”, 
ty ty? ts? 
14336 1 [(arz) (biz) (c12) (dha) (€12) (fiz) (gra) (Are) J] th te 
7168 1 [ (diz) (biz) (C12) (dha) (€12) (fiz) (giz) (hz) _| pos fh ts 
2688 1 [ die A Dye - Cy. dye . E42 Sie - fiz: hy | 
ty ty ts” dig. bie. Ce - fia. ts 
672 is 3 [ die : Dye C12. diy - E19 Sir -Ji2- hi | ty Qj. bis ~ diy = ty 
hy.ts, th fir. di. gi2-t hie. ts, 
ty die. bie. fiz. gio. te ts 
336 1 [1] ti te ts 
[ aye = Dip Cio. dip . €19 fie -Ji2- Ing | ty ts” a2. brs ° hy . ty? 


2188 


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ong bey vow ’ 


